Download Search for Scalar Top Quark Partners and Parton Shower Tuning in

Faculty of Sciences
Department of Physics and Astronomy
Search for Scalar Top Quark
Partners and Parton Shower Tuning
in Modelling Top Quark Pairs
By
Nicolas Bulté
Promotor: Dr. Efe Yazgan
A thesis presented for the degree of
Master of Science in Physics and Astronomy
June 2016
Acknowledgements
To start with, I would like to express my gratitude towards Prof. Dirk Ryckbosch for his
efforts into bringing me towards the point where I finalise my studies. Five years ago,
when I was unsure whether physics was a good fit for me, he told me that all it takes is a
decent dose of enthusiasm and motivation. Bearing this in mind I took on the challenge
and ended up in my final year, writing a thesis within the field of elementary particle
physics. I was given the unique opportunity to stay and work at CERN gaining valuable skills and experience, from which I will be able to profit from for the rest of my life.
Further, I want to thank Prof. Didar Dobur for her endless patience and efforts into
teaching me how experimental particle physics is conducted in practice. Also for the
many hours she spent explaining me difficult concepts, providing me with feedback and
learning me how to perform better as an experimental physicist.
Next, I want to thank Dr. Efe Yazgan for always being there when I was in need of
assistance, for his thorough help, feedback and guidance and for making my research
count. He taught me how scientific research is conducted within an international setting
which made writing this thesis an invaluable experience.
Another big thanks go to Michael Sigamani, Ward Van Driessche and Antoine Pingault
for taking me in their office and allowing me to see the behind the scenes work of scientific research. I’ve had a great time there and I truly enjoyed our talks and laugh bursts.
Finally, I want to thank my family for their support during the past five years. For
keeping up with me through the stressful periods and for providing me with the opportunities to take on this course of study.
1
Contents
1 The Standard Model
1.1 Fundamental Particles . . . . . . . .
1.1.1 The Fermions . . . . . . . . .
1.1.2 The Bosons . . . . . . . . . .
1.2 Fundamental Interactions . . . . . .
1.2.1 The Electromagnetic Force . .
1.2.2 The Weak Force . . . . . . . .
1.2.3 The Strong Force . . . . . . .
1.2.4 Electroweak Unification . . .
1.2.5 Parity and Parity-Violation .
1.3 The Brout-Englert-Higgs-Mechanism
1.4 Limitations of the Standard Model .
2 Supersymmetry
2.1 General introduction . . . . . . . . .
2.2 Supermultiplets . . . . . . . . . . . .
2.3 The Minimal Supersymmetric Model
2.3.1 Particle Categorisation . . . .
2.3.2 Broken Symmetry . . . . . . .
2.3.3 R-parity . . . . . . . . . . . .
2.4 SUSY-searches at the LHC . . . . . .
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24
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3 The CMS Experiment at the LHC
3.1 The Large Hadron Collider . . . . . . . .
3.2 The CMS-Experiment . . . . . . . . . .
3.2.1 The Tracker . . . . . . . . . . . .
3.2.2 The Electromagnetic Calorimeter
3.2.3 The Hadronic Calorimeter . . . .
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3.2.4
3.2.5
3.2.6
The Superconducting Solenoidal Magnet . . . . . . . . . . . . .
The Muon System . . . . . . . . . . . . . . . . . . . . . . . . .
Triggering and Data Acquisition . . . . . . . . . . . . . . . . . .
4 Modelling and Analysis Software
4.1 CMSSW Application Framework . . . . . .
4.2 Particle-flow Event Reconstruction . . . . .
4.3 Monte Carlo Event Generators . . . . . . . .
4.3.1 Powheg . . . . . . . . . . . . . . . .
4.3.2 Pythia . . . . . . . . . . . . . . . . .
4.3.3 Tuning of event-generators: Professor
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5 Analysis Part I: Search for Scalar Top Quark Partners
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 MT 2 (ll) . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Analysis strategy . . . . . . . . . . . . . . . . . . . . . .
5.4 MT 2 (ll)-tail events study for the tt̄ + jets background . .
5.5 Reduction of the tt̄ + jets background . . . . . . . . . .
5.6 Summary and Discussion . . . . . . . . . . . . . . . . . .
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6 Analysis Part II: Parton Shower Tuning in Modelling Top Quark Pairs
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Tuning of αSISR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Estimation of uncertainty on αSISR . . . . . . . . . . . . . . . . . . . . .
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 The renormalisation scale µR . . . . . . . . . . . . . . . . . . . . . . .
6.6 Comparison between Professor Interpolation and MC Output . . . . . .
6.7 Validation with other Top Quark Distributions . . . . . . . . . . . . . .
6.8 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
63
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7 Nederlandstalige Samenvatting
7.1 Introductie . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Deel I: Zoeken naar scalaire top quark partners . . . . .
7.3 Deel II: Scherpstellen van αSISR in POWHEG+PYTHIA8
delleren van top quark paren . . . . . . . . . . . . . . . .
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. . .
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voor
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het mo. . . . .
A Professor: A tuning tool for Monte Carlo event generators
A.1 Determining the response function MCb (p) . . . . . . . . . . . . . . . .
A.2 Goodness of Fit (GoF) . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
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84
85
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A.3 Error Estimation: Construction of Eigentunes . . . . . . . . . . . . . .
4
86
Chapter 1
The Standard Model
This chapter provides a general outline of the general principles behind the Standard
Model (SM), one of the currently most successful theories within the field of physics.
The SM is a theory concerning the 3 fundamental interactions of the electromagnetic,
weak and strong forces and classifies all subatomic particles known today. The SM
explains a wide range of experimental results, stemming from a combined effort of
scientists worldwide during the second half of the 20th century. Despite its power to
explain and predict, it is incomplete. The shortcomings of the SM will be discussed at
the end of this chapter.
1.1
Fundamental Particles
The fundamental particles are divided in 2 groups: the fermions and the bosons, shortly
introduced in the following sections.
1.1.1
The Fermions
The first class of particles discussed, the fermions, have spin = 1/2. They are grouped
in 3 families according to their masses. Each fermion has a corresponding antifermion.
The fermions, in turn, are grouped into 2 classes, the leptons and the quarks, summarised together with some of their properties in Table 1.1. The 6 leptons are the
electron (e), muon (µ) and tau-lepton (τ ) with corresponding electron- (νe ), mu- (νµ )
and tau-neutrinos (ντ ). The 6 quarks are the up (u), down (d), charm (c), strange (s),
top (t) and bottom-quark (b).
5
Quarks carry apart from electric
charge, also a colour charge. Experimentally it is observed that
no free colour states exist in nature. This means that quarks
(nor gluons, to be explained
later) can propagate freely and
cannot be observed singularly.
Quarks can be bound together
forming so-called hadrons (protons, neutrons, pions, ...). The
process of forming hadrons is
called hadronisation. Leptons
on the other hand do not carry
colour charge and can certainly
be found to propagate freely.
1.1.2
Table 1.1: Tabular overview of the fermions comprised within the Standard Model. The subscripts L
and R stand for the left- and right handed components, T is the weak isospin and T3 its third component [1].
The Bosons
Apart from gravitation (which is not comprised within the Standard Model), 3 other
fundamental interactions are known. Each of those is mediated by the exchange of
one or more vector bosons, which are spin = 1 particles. They are together with their
exchange bosons listed below:
• The strong interaction: 8 gluons (g) coupling to colour charge.
• The electromagnetic interaction: the photon (γ) coupling to electric charge.
• The weak interaction: the W ± -bosons and the Z 0 -boson coupling to weak
charge.
As can be seen from Table 1.1, the quarks carry colour-charge, allowing them to couple
to themselves. The W bosons and Z bosons can also couple to themselves, as they carry
weak charge. The photon does not share this property as it doesn’t carry electric charge.
The only scalar particle, the Higgs-boson, requires some special attention and will be
discussed in section 1.3 concerning the Brout-Englert-Higgs Mechanism.
6
1.2
Fundamental Interactions
The SM is described by a Lagrangian, which describes the whole dynamics of the system. The symmetry group representing the SM Lagrangian is SU(3)×SU(2)×U(1). A
symmetry group defines under which transformations a theory is invariant. According
to Noether’s theorem a conservation law is associated to a physical system for each
continuous symmetry of its action [2]. For the SM, the conserved quantities are colour
SU(3), weak isospin SU(2) and electric charge U(1).
As was discussed in Sec. 1.1.2, the (gauge-)bosons mediate the SM interactions. In
general, gauge-bosons appear as virtual particles between two interaction vertices (as
can be schematically depicted by Feynman-diagrams). Depending on which interaction
(and the energy-scale at which the interaction occurs) a coupling strength is associated,
denoted by α. In this section, each of those fundamental interactions will be discussed
in more detail.
1.2.1
The Electromagnetic Force
The electromagnetic force is the one that acts between electrically charged particles and is fully
described by the relativistic quantum field theory of quantum electrodynamics (QED), with
abelian symmetry group U(1). The gauge-boson
mediating the electomagnetic interaction is the
massless and electrically neutral photon γ, allowFigure 1.1: A QED process.
ing the interactions up to an infinite range (the
forces between particles grow weaker with the distance between them). The photon can interact with every particle carrying an electric
charge Q (as can be seen from Fig. 1.1 in which two electrons with Q = -1e interact,
exchanging a photon). The coupling strength of the electromagnetic interaction has the
1
[3]. Thanks to the coupling being smaller than 1, this invalue of αem (Q2 = 0) ≈ 137
teraction allows calculations in perturbation theory. It needs to be noted still that αem
is a running coupling constant, meaning it depends on the value of transferred momentum Q2 . This is a consequence of the fact that the photon isn’t capable of undergoing
self-interaction, as it doesn’t carry any electric charge. This can be understood with
the picture that every charged particle is surrounded by a cloud of virtual γ’s and e+ e− pairs which result in a charge screening effect. The higher the transferred momentum,
the closer one probes the target under study and the fewer charge screening remains.
This results in a higher value of αem .
7
1.2.2
The Weak Force
The weak interaction is mediated by 3 gauge
bosons: the electrically neutral Z 0 boson with
mass MZ 0 = 91.2 GeV/c2 , and the electrically charged W + and W − bosons with MW ±
= 80.4 GeV/c2 . Due to the large masses of
the exchange bosons, the weak interaction has
a very short range. This allowed Enrico Fermi
in the 1940s to develop a theory of interaction described as a point interaction for weak
Figure 1.2: A weak interaction pro- processes. This was implemented in a quancess.
tum field theory based on a SU(2) symmetry.
The quantum number that governs the weakly interacting particles is the weak isospin.
It is the third component, T3 , that becomes relevant because it is conserved between initial and final states. The left-handed fermions have non-zero T3 : the up-type fermions,
have T3 = + 21 and all down-type fermions have T3 = − 21 .1 All right-handed fermions
have T3 = 0. If one assigns T3 (W ± ) = ±1, then from Fig. 1.2 one can see that weak
isospin is indeed conserved. The coupling strength of the weak interaction is 4 to 5
orders of magnitude smaller than αem at low momentum transfers.
Another important artifact of the weak interaction is the following. Neutral currents
through Z 0 -exchange couple between quarks of the same flavour whereas charged currents through W ± -exchange allow coupling between different generations. For leptons,
both neutral and charged currents couple to particles inside the same generation. The
behaviour for quarks can be understood in terms of the mixing mechanism. When a
quark is produced as a mass eigenstate in a certain generation, it decays via the weak
interaction as a weak eigenstate which is a superposition of quarks of all three generations. Mathematically the mixing mechanism is described by the Cabibbo-KobayashiMaskawa (CKM) matrix, which contains the weak eigenstates in terms of the mass
eigenstates, as can be seen from Eq. 1.1.
 0
  
  
d
d
Vud Vus Vub
d
s0  = VCKM · s =  Vcd Vcs Vcb  · s
(1.1)
0
t
t
Vtd Vts Vtb
t
1
Anti-fermions get the same T3 -value as their fermion counterparts.
8
An interesting type of fermions are the neutrinos (see Sec. 1.1.1) as they only interact through the weak force and gravity2 , but are not effected by electromagnetic nor
strong forces. Their existence was postulated by Wolfgang Pauli in 1930 to explain the
occurrence of β-decay
n0 → p+ + e− + ν̄e .
(1.2)
The combined energies of the proton p+ and electron e− do not add up to the energy of
the neutron n0 . Therefore Pauli proposed the existence of an electrically neutral neutrino νe in order for energy conservation to be satisfied. Note that also spin conservation
is now satisfied, where the anti-neutrino is right-handed and the electron left-handed3 .
In 1956, β-capture was proposed to detect neutrinos. Antineutrinos created in a nuclear reactor from β-decay interact with protons according to ν̄e + p+ → n0 + e+ . The
positron directly annihilates with an electron forming two photons and the neutron
can interact with a nucleus, also releasing a photon. This signature is unique for an
antineutrino interaction [5]. Later, with the discoveries of the muon and tau-lepton, it
was known that neutrinos come in three flavours (νe , νµ , ντ ).
Neutrinos can be produced in various ways apart from nuclear reactors. Most neutrinos
on Earth originate from nuclear processes inside the Sun (pp chain, 7 Be, 8 B, etc. [6]).
Another very intensive source of neutrinos is a supernova which was measured from
the SN 1987A supernova in 1987. It is also thought that a large amount of neutrinos
must be present in the Cosmic Neutrino Background (CνB), a remnant from the Big
Bang. Due to their incredibly low energies however, the CνB has not yet been observed.
Neutrino-experiments (e.g. IceCube, Soli∂, ...) are widely conducted around the globe:
they provide gateways to understand more about the cosmos (from the structure of
the universe to dark matter) and provide opportunities to acquire a more thorough
understanding about physics beyond the standard model (neutrino-oscillations, reactor
anomaly, etc.).
2
Neutrinos were recently proven to have a small mass, see the paragraph in Sec. 1.4 on neutrinooscillations, a phenomenon not predicted by the SM.
3
A right-handed particle has its spin in the direction of its momentum whereas a left-handed particle
has its spin in the opposite direction to its momentum. One often also speaks of respectively helicity
h = +1 and h = −1. The helicity of the neutrino was determined by Goldhaber in 1958 [4].
9
1.2.3
The Strong Force
The strong force acts between particles carrying
colour charge and is described in the framework of
quantum chromodynamics (QCD) with nonabelian symmetry group SU(3). The mediating
bosons are called the gluons that have spin 1 and
couple to colour charge, a conserved quantity for
the strong interaction. The 3 possible types of
colours are green (g), red (r) and blue (b) and
Figure 1.3: A QCD process.
their respective anticolours. Each type of quark
and gluon carry a non-zero colour charge. A quark
carries one of 3 colours whereas a gluon carries a colour-anticolour combination. In total there are 8 gluons.
Gluons carry colour charge themselves, allowing them to emit other gluons. This property, which is absent in QED, yields the following picture. When the momentum transfer
Q2 increases, gluons start to fluctuate more and more in secondary gluons and cause an
anti-screening effect [1]. Because of this, at close distances, the gluon sees less colour
charge. This means that the coupling strength grows weaker for higher energy-scales
that explains the running of αs .
In the limit Q2 → ∞, quarks can be considered to be free due to the very low interquark
coupling. This limit is called asymptotic freedom. At large distances however, the
interquark coupling increases so strongy that it’s impossible to detach quarks from
hadrons, this principle is called confinement and can be understood from a fluxtube
being formed between two particles holding colour charge (Fig. 1.4).
Figure 1.4: Left: Electromagnetic field lines for two separated electrically charged
particles. The further apart the particles go, the weaker the field gets (V (r) ∼ 1/r).
Right: Strong interaction field lines in the form of a flux tube. The further apart the
particles go the stronger the field gets (V (r) ∼ r) [1].
10
Confinement originates from the potential to rise as V (r) ∼ r which means that the
longer the fluxtube gets, the more energy is stored in the field. Eventually the fluxtube
breaks and creates new q q̄ pairs.
From renormalisation group theory in QCD [7], it follows that the strong coupling
constant αS at a particular scale µ1 can be written into an equation in which it is
linked to itself at another scale µ2
αS (µ21 ) =
αS (µ22 )
2
µ1
µ22
1 − αS (µ22 )β1 ln
(1.3)
With β1 given by
11nc − 2nf
(1.4)
12π
in which nc = 3 denotes the possible number of colour charges and nf the number of
active quark flavours (this depends on the chosen scale). Here, only one loop is taken
into account as can be seen from the computation of β1 in the renormalisation group
equation (with aS = απS )
β1 =
das
= β1 a2s + β2 a3s + ...
(1.5)
dµ
It is this equation that for one loop corrections can be analytically transformed into
Eq. 1.3. If one now chooses the scale µ2 = ΛQCD such that Eq. 1.3 becomes infinite
(or the denominator becomes 0) we find the following relationship
β(as ) = −µ
ΛQCD = µ1 e
−
1
2β1 αS (µ2
1)
(1.6)
That way, Eq. 1.3 can be rewritten as
αS (µ2 ) =
β1 ln
1
µ2
(1.7)
Λ2QCD
The constant ΛQCD is the scale at which perturbation theory breaks down. Asymptotic
freedom is also evident from the above equation, for µ → ∞ the coupling constant αS
vanishes.
An overview of different measurements of αS at the scale of the mass of the Z boson
MZ is given in Fig. 1.5. The current world average is αS (MZ2 ) = 0.1184 ± 0.0007 [8].
11
36
9. Quantum chromodynamics
τ-decays
Baikov
Davier
Pich
Boito
SM review
structure
e+e- annihilation
functions
ABM
BBG
JR
NNPDF
MMHT
lattice
HPQCD (Wilson loops)
HPQCD (c-c correlators)
Maltmann (Wilson loops)
JLQCD (Adler functions)
PACS-CS (vac. pol. fctns.)
ETM (ghost-gluon vertex)
BBGPSV (static energy)
ALEPH (jets&shapes)
OPAL(j&s)
JADE(j&s)
Dissertori (3j)
JADE (3j)
DW (T)
Abbate (T)
Gehrm. (T)
Hoang
(C)
electroweak
precision fits
hadron
collider
GFitter
CMS
(tt cross section)
Figure
9.2: Summary
determinations
of αs (MZ2 ) offrom
the
sub-fields
Figure
1.5: Summary
of the of
results
various measurements
αS (M
world
Z ). sixThe
2
discussed
text. The
yellow
shaded)
and dashed
linesand
indicate
average
αS (MZin) the
= 0.1184
± 0.0007
is (light
indicated
withbands
the black
dotted line
grey the
pre-average
uncertainty
bandsvalues
[9]. of each sub-field. The dotted line and grey (dark shaded) band
represent the final world average value of αs (MZ2 ).
12
whereby the dominating contributions to the overall error are experimental (+0.0017
−0.0018 ), from
+0.0013
parton density functions (−0.0011 ) and the value of the top quark pole mass (±0.0013).
February 10, 2016
16:30
With the concepts of asymptotic freedom and confinement introduced, for calculations
one cannot ignore the very important principles of factorisation. The general problem
addressed by factorisation is how to calculate high-energy cross sections [10]. A general
cross section contains both short range ( Q2 ) and long range ( Q2 ) contributions.
The long range part cannot be calculated perturbatively in QCD. Factorisation allows
us to derive a prediction for those cross sections by separating long from short range
behaviour in a systematic way. A general cross section, e.g. σpp→tt̄ , can be written as
Z 1
X Z 1
dx2 fiPDF (x1 , µ2F )fjPDF (x2 , µ2F )σij→tt̄ (µ2R )
(1.8)
dx1
σpp→tt̄ =
i,j=partons
0
0
It consists of summing over all possible partons inside the protons that can give rise
to the hard scattering process, and integrating over the momentum fractions x1,2 . The
integrandum consists of two Parton Density Functions (PDFs) for each of the two interacting partons i and j. They represent the probability of finding parton i or j carrying
a momentum fraction xi,j of the proton, when the proton is probed at a scale µF .
The PDFs are non-perturbative, process-independent functions that can be obtained
from various measurements. To find the PDF at other scales, one can be derived from
the other using the DGLAP evolution functions [11]. The integrandum also contains
the cross section that only takes the hard scattering into account between both partons
and can be written in a perturbative expansion in the running coupling constant αS [12].
Measuring the tt̄ cross-section provides good tests of perturbative QCD and our understanding of the underlying (perturbative) hard scatter cross section. For this however,
we need to have reliable PDF sets f PDF (x, Q2 ). Those are acquired from measurements
in deep inelastic electron, muon, neutrino and antineutrino scattering experiments and
collider data. Dedicated groups have constructed various PDF sets, e.g. NNPDF [13],
CTEQ5 [14], CT10, [14], etc.
1.2.4
Electroweak Unification
In the 19th century, Maxwell developed a formalism in which both electricity and magnetism can be described by one framework called electromagnetism. A similar unification was done in the 1960s by Weinberg, Salam and Glashow who combined electromagnetism and the weak interaction into one picture. This electroweak unification
will be explained in this section.
13
Earlier was mentioned how the third component of weak isospin T3 of the W + and W −
bosons are +1 and -1 respectively. A third state should therefore exist with T = 1,
T3 = 0. This state is denoted by W 0 and forms together with W + and W − the weak
isospin triplet. One further postulates the existence of a singlet state B 0 with T = 0,
T3 = 0. The W 0 nor the B 0 are found in nature, they are mathematical constructs
that form two other known neutral gauge bosons: the photon and the Z 0 . The idea
of electroweak unification is to describe the photon and Z 0 as two orthogonal linear
combinations of the W 0 and B 0 . This is expressed through the so-called electroweak
mixing angle θW :
|γi = cos(θW ) B 0 + sin(θW ) W 0
0
Z = −sin(θW ) B 0 + cos(θW ) W 0
If we take a look at the dominant Z 0 boson decay modes [8]:
Z 0 → e+ e−
µ+ µ−
τ +τ −
ν̄e,µ,τ νe,µ,τ
(3.363 ± 0.004)%
(3.366 ± 0.007)%
(3.370 ± 0.008)%
(20.00 ± 0.06)%
we can see how the decay probability into charged leptons is significantly different from
the decay probability to neutrinos. Hence the Z 0 cannot just be a neutral W 0 state
with equal couplings to all fermions. Its coupling clearly depends on the electric charge,
and this is a direct consequence of the fact that the Z 0 has a small electromagnetic
component.
14
1.2.5
Parity and Parity-Violation
An important quantity in subatomic physics is parity P. A parity transformation is
defined by
 
 
x
−x
P : y  7→ −y 
(1.9)
z
−z
in which the spatial coordinates x, y and z flip their signs. It is a quantity that is
conserved by electromagnetism, the strong interaction and gravity, but it is violated
by the weak interaction. Parity (or P-) violation was proposed by T.-D. Lee and C.-N.
Yang and experimentally proven by the Wu experiment conducted by C.S. Wu.
~ field nuclei were used. These undergo
In the Wu experiment, polarised 60 Co in a B
a three stage decay process; namely a nuclear β-decay followed by two high energy
photons according to
60
Co(5+ ) →
60
Ni∗∗ (4+ ) →
60
Ni∗∗ (2+ ) →
Ni∗∗ (4+ ) + e− + ν̄e
60
Ni∗∗ (2+ ) + γ
60
Ni∗∗ (0+ ) + γ
60
Weak
Electromagnetic
Electromagnetic
the numbers between parentheses denote the spin-parity J P . In the experiment, photons are counted at angles 0◦ and 90◦ with respect to the principal axis along the
polarisation direction. The photon emission is anisotropic and this anisotropy serves
as a measurement of the polarisation: γ = (W (90◦ ) − W (0◦ ))/(W (90◦ ) + W (0◦ )). If
γ = 0, then the nuclei have lost all polarisation. The electron counting rate We (θ) is
measured and should be equal if parity is conserved by weak interactions for We (0◦ )
~ field.
and We (180◦ ). Measuring this can be achieved by flipping the B
The observation by Wu et al. [15] was that electrons are more likely to be emitted opposite to the field than along it, proving that parity is violated by the weak interaction.
Moreover, it seems that parity is maximally violated, leading to the V − A interaction
theory proposed by Feynman and Gell-Mann [16].
15
1.3
The Brout-Englert-Higgs-Mechanism
The Brout-Englert-Higgs-Mechanism was named after the three physicists Robert Brout,
François Englert and Peter Higgs, based on their work in the 1960s [17]. This mechanism is essential to explain the generation mechanism of mass for the W and Z gauge
bosons, while the photon remaining massless. Furthermore, it predicts the existence of
a massive scalar, spin-0 particle: the Higgs boson. As the Higgs boson was eventually
discovered at the LHC by the CMS and ATLAS experiments in 2012, it deserves a
deeper study of it’s theoretical backgrounds.
Out of symmetry considerations, the weak force’s gauge bosons should have zero mass,
which clearly is not the case. This conundrum is fixed through introduction of a quantum field, the Higgs field, which stretches throughout all space. The masses of the W
and Z bosons will arise from breaking the SU(2)L × U(1)Y local gauge symmetry of
the electroweak sector of the Standard Model. This will be more elaborately discussed
below, where most of the discussion is based on [18].
Consider a scalar field φ with the potential
1
1
V (φ) = µ2 φ2 + λφ4
2
4
with the corresponding Lagrangian
(1.10)
1
1
1
L = (∂µ φ)(∂ µ φ) − µ2 φ2 − λφ4 .
(1.11)
2
2
4
1
µφ2 represents the mass of the particle, the 41 λφ4 represents a quartic coupling of the
2
field φ with coupling strength 14 λ. The vacuum state is the lowest energy state of the
field φ and corresponds to the minimum of the potential. It has its minima at
r
−µ2 φ = ±v = ± (1.12)
.
λ If one considers µ2 < 0, the field has a non-zero vacuum expectation value. There
are two possible vacuum states: φ = +v and φ = −v. The choice of one of the
vacuum states breaks the symmetry of the Lagrangian, this is called spontaneous
symmetry breaking. If one makes an arbitrary choice of the vacuum state to be
φ = +v, small perturbations of the field around the vacuum state can be considered by
writing φ(x) = v + η(x). The Lagrangian now becomes
16
1
1
1
L(η) = (∂µ η)(∂ µ η) − µ2 (v + η)2 − λ(v + η)4 .
(1.13)
2
2
4
With the minimum of the potential given by µ2 = −λv 2 , the expression can be written
as
1
1
1
L(η) = (∂µ η)(∂ µ η) − λv 2 η 2 − λvη 2 − λη 4 + λv 4 .
2
4
4
Comparing this with the Klein-Gordon equation for spin-0 scalar particles
1
L = (∂µ φ)(∂ µ φ) −
2
it becomes clear that the term proportional to η 2
mη =
√
2λv 2 =
1 2 2
mφ
2
represents a mass
(1.14)
(1.15)
p
−2µ2
and therefore the Lagrangian 1.15 in terms of the excitations about the minimum describes a massive scalar field. If we apply the same principles of spontaneous symmetry
breaking to a complex scalar field, more interesting properties emerge. With a field
1
φ = √ (φ1 − iφ2 )
2
(1.16)
the Lagrangian becomes
1
L = (∂µ φ)∗ (∂ µ φ) − µ2 (φ∗ φ) + λ(φ∗ φ)2
2
1
1
1
1
= (∂µ φ1 )(∂ µ φ1 ) + (∂µ φ2 )(∂ µ φ2 ) − µ2 (φ21 + φ22 ) − λ(φ21 + φ22 )2 .
2
2
2
4
(1.17)
(1.18)
This Lagrangian is invariant under φ → φ0 = eiα φ and therefore possesses a global
U(1) symmetry. Depending on the sign of µ2 , the potential has a different shape. For
µ2 < 0 it has an infinite set of minima denoted by the dashed circle line in Fig. 1.6 (b).
Nature will choose one of the vacuum states, causing to spontaneously break the U(1)
symmetry. If one chooses a vacuum state (φ1 ,φ2 ) = (v,0) and expands the scalar field
φ as φ1 (x) = η(x) + v and φ2 (x) = ξ(x) one gets for φ and the Lagrangian L
1
φ = √ (η + v + iξ)
2
1
1
L = (∂µ η)(∂ µ η) + (∂µ ξ)(∂ µ ξ) − V (η, ξ)
2
2
17
(1.19)
(1.20)
Figure 1.6: The potential V (φ) = µ2 (φ∗ φ) + λ(φ∗ φ)2 for two different values of µ2 : (a)
µ2 > 0 and (b) µ2 < 0.
with
1
1
1
1
(1.21)
V (η, ξ) = − λv 4 + λv 2 η 2 + λvη 3 + λη 4 + λξ 4 + λvηξ 2 + λη 2 ξ 2
4
4
4
2
Every quadratic term in the field can be interpreted as a mass whereas all higher order
terms are interaction terms. This allows the Lagrangian to be rewritten as
1
1
1
L = (∂µ η)(∂ µ η) − m2η η 2 + (∂µ ξ)(∂ µ ξ) − Vint (η, ξ).
(1.22)
2
2
2
√
This Lagrangian represents a scalar field η with mass mη = 2λv 2 and a massless
scalar field ξ. This can be understood by looking at the potential in Fig. 1.6 (b), in
which it’s clear that the excitations η are up the potential walls and those of ξ are along
the dashed circle where the potential doesn’t change. The massless scalar particle ξ is
known as the Goldstone boson.
For the complex scalar field, the Lagrangian is not invariant under a local U(1) gauge
transformation
φ(x) → φ0 (x) = eigχ(x) φ(x).
(1.23)
This can however be solved by introducing the replacement of the derivatives in the
Lagrangian by the corresponding covariant derivatives
18
∂µ → Dµ = ∂µ + igBµ .
(1.24)
L = (Dµ φ)∗ (Dµ φ) − V (φ2 )
(1.25)
Then the Lagrangian,
is gauge invariant if the new gauge field Bµ transforms as
Bµ → Bµ0 = Bµ − ∂µ χ(x)
(1.26)
F µν = ∂ µ B ν − ∂ ν B µ
(1.27)
If we take into account that
We find after plugging Eq. 1.20 into the Lagrangian 1.25, that
1
1
1
1
L = (∂µ η)(∂ µ η) + (∂µ ξ)(∂ µ ξ) − Fµν F µν + g 2 v 2 Bµ B µ − Vint + gvBµ (∂ µ ξ) (1.28)
2
2
4
2
There is a problem with the last term in the above expression, which is a direct coupling
between the spin-1 gauge field B and the Goldstone field ξ. This can be eliminated by
performing the gauge transformation
Bµ (x) → Bµ0 (x) +
1
∂µ ξ(x).
gv
(1.29)
The Lagrangian becomes
1
1
1
L = (∂µ η)(∂ µ η) − λv 2 η 2 − Fµν F µν + g 2 v 2 Bµ0 B 0µ −Vint
|2
{z
} |4
{z2
}
massive η
(1.30)
massive gauge field
We notice that the Goldstone field ξ has disappeared. The original gauge transformation
of φ(x) becomes
φ(x) → φ0 (x) = e−ig
ξ(x)
gv
φ(x) = e−iξ(x)/v φ(x).
(1.31)
In first order, the form φ(x) = √1 [v + η(x)] eiξ(x)/v can be rewritten as
(2)
1
φ(x) ≈ √ [v + η(x)] eiξ(x)/v
2
This yields
19
(1.32)
1
1
φ(x) → φ0 (x) = √ e−iξ(x)/v [v + η(x)] eiξ(x)/v = √ (v + η(x)).
2
2
Choosing the field φ(x) to be entirely real, i.e.
(1.33)
1
1
φ(x) = √ (v + η(x)) ≡ √ (v + h(x))
(1.34)
2
2
corresponds to the choosing the previously discussed gauge; the unitary gauge. We
now notice that the only physical field in the unitary gauge is the remaining Higgs-field
h(x). The Goldstone field ξ(x) was “eaten” by the gauge field. If we write µ2 = −λv 2 ,
the Lagrangian can be finally rewritten as
1
1
1
1
L = (∂µ h)(∂ µ h) − λv 2 h2 − Fµν F µν + g 2 v 2 Bµ B µ + g 2 vBµ B µ h2 − λvh3 − λh4 .
{z
} |
|
|2
{z
} |4
{z2
} interactions
{z 4 }
h, B
massive h
massive gauge boson
self-interaction h
(1.35)
This Lagrangian describes a massive scalar Higgs field h and a massive gauge boson B,
obeying a U(1) local gauge symmetry. We see thus how the Higgs-mechanism effectively
generates mass for the gauge bosons and predicts the existence of a massive scalar
particle, the Higgs Boson. In the Salam-Weinberg model, the mass terms are found by
writing the Lagrangian such that it respects the SU (2)L × U (1)Y local gauge symmetry
of the electroweak theory by replacing the derivatives by the appropriate covariant
derivatives. We omit the calculation and give the results below. The mass terms for
the W (1) and W (2) spin-1 fields appear as:
1
1
1
mW Wµ(1) W (1)µ
and
mW Wµ(2) W (2)µ with
mW = gW v
(1.36)
2
2
2
The W (3) - and B-terms couple together in the Lagrangian and mix. The relationship
between the physical fields and the underlying ones in the Lagrangian is
Aµ = cosθW Bµ + sinθW Wµ(3)
with mA = 0
q
1
2
with mZ = v gW
Zµ = −sinθW Bµ +
+ g 02
2
Where Aµ represents the massless photon, and Zµ the massive Z-boson. The BroutEnglert-Higgs-Mechanism thus effectively predicts the masses of the gauge bosons, as
well as the existence of a massive scalar spin-0 boson. After the discovery of the Higgs
boson in 2012 at the CERN LHC, François Englert and Peter Higgs were awarded the
2013 Nobel Prize in physics.
cosθW Wµ(3)
20
The Brout-Englert-Higgs mechanism can be used to generate the masses for the fermions.
Without giving the exact theoretical considerations, a new interaction, the Yukawa interaction, is introduced that provides the coupling between a scalar field φ and a Dirac
field ψ. Dirac fields describe the fermions with spin-1/2. The Yukawa couplings of the
fermions to the Higgs field are given by
√ mf
(1.37)
2
v
with v = 246 GeV the vacuum expectation value of the Higgs field. They occur in a
Lagrangian of the form
gf =
mf ¯
L = −mf f¯f −
ffh
v
1.4
(1.38)
Limitations of the Standard Model
Apart from the great successes of the Standard Model, some fundamental problems still
go unexplained. In essence, it is still unsure how the SM behaves at very high energy
scales. As accelerators grow stronger, it might become possible to hit up till now unexplored frontiers and new physics start emerging. An overview of the most important
indicators that the SM is incomplete, and the challenges it poses for physicists today,
is listed below.
General relativity
The Standard Model does not incorporate the fourth fundamental force: the gravitational interaction. In fact, the SM has to become compatible with general relativity,
and this is not the case.
Neutrino-oscillations
In the early 2000’s the Super-Kamiokande Collaboration [19] and Sudbury Neutrino Observatory (SNO) [20] Collaborations confirmed the occurrence of neutrino-oscillations:
the changing of neutrino flavour as neutrinos travel a particular distance. In case of two
neutrino generations, the probability of a neutrino να changing flavour to νβ is given
by
∆m2 L
2
2
Pα→β = sin (2θ) sin
(1.39)
4E
with θ the mixing angle between the neutrino flavour- and mass-eigenstates, E its energy, L its distance travelled and ∆m its mass difference. This mechanism thus proves
21
that neutrinos must have a (small) mass. This phenomenon is not predicted by the
Standard Model.
The Hierarchy problem
The so-called hierarchy problem boils down to the question why the weak force is 1032
times stronger than gravity. In another formulation, one can ask why the Higgs mass
is so small as was experimentally found [21], [22]. The Higgs boson has a bare mass4
that gets corrected for each particle interacting with it [23]. The heavier the particle,
the larger this correction. The top quark adds such a large correction that it is at least
peculiar why the bare Higgs mass is so different from the measured one.
Figure 1.7: Rotational curve of an arbitrary
galaxy. A: model without dark matter. B:
model with dark matter, as it is measured.
Figure 1.8: Assumed energy distribution of
the universe.
Dark Matter and Dark Energy
Dark matter and dark energy are hypothetical substances believed to account for over
96% of all energy in the universe (Fig. 1.8). Dark matter has never been directly
observed, as it does not interact electromagnetically, but its existence is inferred from
various gravitational effects. One of the most important reasons to believe that there
is such a thing as dark matter is that it is a necessary ingredient to explain the observationally measured rotation curves of galaxies, as depicted in Fig. 1.7. It is observed
that at large distances from the galaxy center, stars move at larger rotational speeds
4
The bare mass is the mass of the particle when the scale of distance to it approaches zero. The
measured mass is the “dressed” mass, which is an addition to the bare mass because the particle
interacts with virtual particle pairs temporarily created by the force field around the particle.
22
than can be explained from the observed mass distribution. Therefore extra matter
is incorporated in the model: dark matter. Another example are the various gravitational lenses that have been observed. Those are massive objects (typically galaxies)
bending the trajectories of light coming from behind and distorting the images seen
through telescopes. The lens is often observed to be much less heavy than the lensing
effect it causes, adding to the belief there is more mass present than can be observed.
Other evidence can be found in measured anisotropies in the angular distribution of
the Cosmic Microwave Background (CMB) in which cold dark matter is introduced as
an explanation of this anisotropy5 . The Standard Model does not contain any particle
candidate that could be explained as being the primary dark matter component, it thus
only explains ∼4% of observable matter in the universe. There are models that incorporate DM particles into the SM (e.g. MACROs [25]), however those are less popular.
Matter-antimatter asymmetry
The observable universe is made of primarily matter and contains only small proportions of antimatter. The Standard Model predicts that matter and antimatter should
have been created in (almost) equal amounts when the universe was formed. There is
no mechanism incorporated within the SM that can explain the observed asymmetry.
All predictions made by the SM have been confirmed, often with high precision. At the
time of writing there is no single theory that successfully incorporates all the needed
extensions explained above (especially general relativity, which is also highly successful).
The most popular extension is Supersymmetry (SUSY) and is very intensively searched
for at the LHC. A general introduction to SUSY is given in Chapter 2.
5
This is often called the ΛCDM model. Recent CMB anisotropies were measured by WMAP [24]
23
Chapter 2
Supersymmetry
2.1
General introduction
As is alluded in the previous chapter, the Standard Model clearly is an incomplete
theory. Up till now, no framework exists to describe the physics at very high energies
up to the Planck scale, with MP = (8πGNewton )−1/2 = 2.4×1018 GeV/c2 , where quantum
gravitational effects become important. This was already addressed before in a brief
mentioning of the so-called hierarchy problem (Sec. 1.4). The Higgs boson receives
quantum corrections to its mass originating from the virtual effects of particles that
couple to the Higgs field. This is described in more detail in Ref. [26], where a large part
of the discussion in this chapter is based on. Consider for example the loop diagram
given in Fig. 2.1. In the Lagrangian a term of the form -λf H f¯f will originate and
yields a correction (after applying the Feynman rules, which is omitted here)
|λf |2 2
Λ + ...
(2.1)
8π 2 UV
with λf the Yukawa-coupling. The mass of a fermion is proportional to its Yukawacoupling meaning that the Higgs boson couples most strongly to the most massive
particles.
∆m2H = −
Figure 2.1: One-loop quantum correction to the Higgs squared mass parameter m2H .
24
The negative sign originates from the spin-statistics theorem, which dictates that fermions
yield negative contributions whereas bosons yield positive contributions. The factor
ΛUV is called the ultraviolet cutoff and denotes the scale up to which the Standard
Model is valid. Filling the top quark into the equation yields unnaturally high corrections to MH2 at the Planck mass, since the correction depends quadratically on the
cut-off. This particular problem can be solved by the principles of supersymmetry.
It is an extension of the SM that creates superpartners for all Standard Model particles.
In our particular example, the addition of supersymmetric partners adds extra terms to
Eq. 2.1 and yields cancellation between them, effectively reducing the correction to zero.
Formally, Supersymmetry or SUSY, is a symmetry that relates bosons and fermions.
A supersymmetry transformation turns a bosonic into a fermionic state and vice versa.
This is done through the introduction of an anti-commuting spinor with
Q |Bosoni = |Fermioni
Q |Fermioni = |Bosoni
2.2
(2.2)
Supermultiplets
The supersymmetric particles are divided in supermultiplets . Each of those contain both boson and fermion states who are superpartners of each other. Because the
squared-mass operator -P 2 commutes with the operators Q and Q† it follows that particles within the same supermultiplet have the same masses. The generators Q and
Q† also commute with the generators of the gauge transformations. This has the consequence that particles within the same supermultiplet must have the same electric
charges, weak isospin and color charges. A final very important remark is that it can
be proven from quantum mechanical considerations (done in [26]) that the number of
degrees of freedom for fermions and bosons must be exactly equal within the same supermultiplet. Therefore one has
nF = nB .
(2.3)
This property can then be used to construct two types of multiplets that are used to
classify the supersymmetric particles.
25
Chiral supermultiplets
The first considered supermultiplet consists of a single Weyl fermion (with two spin
helicity states; nF = 2) and two real scalars (nB = 1). The two real scalar degrees
of freedom are then combined into a complex scalar field. The combination of a twocomponent Weyl fermion and a complex scalar field is called a chiral supermultiplet.
Gauge supermultiplets
A second possible supermultiplet contains a spin-1 vector boson that is massless (in order for the theory to be renormalizable). Such a boson has 2 helicity states which yields
nB = 2. Its superpartner must therefore have two degrees of freedom, nF = 2, and this
is again a massless spin-1/2 Weyl fermion. The combination of spin-1/2 gauginos and
spin-1 gauge bosons is called a gauge supermultiplet.
In a supersymmetric extension of the SM, each of the already known fundamental particles are classified into a chiral or gauge supermultiplet. They each have a superpartner
that has a spin differing by 1/2 unit. Other combinations of spins are possible but those
can always be reduced to chiral or gauge supermultiplets [26].
2.3
The Minimal Supersymmetric Model
The Minimal Supersymmetric Model (MSSM) is a minimal supersymmetric extension
of the Standard Model. It contains the least number of particles you can add to the
Standard Model to make a viable SUSY model. A first step is to categorise the already
known SM particles into the SUSY picture, then supersymmetry breaking will be discussed, and a conserved quantity called R-parity will be introduced. This section is
based on Ref. [26].
2.3.1
Particle Categorisation
The Standard Model fermions (quarks and leptons) are members of chiral supermultiplets. This is because left-handed fermions couple differently under the gauge group
than their right-handed parts. The supersymmetric particles of fermions are spin-0
particles. As a naming convention, the letter “s” is put in front of the SM partner’s
names (“s” denotes “scalar”). This way, the sfermions are subdivided into squarks and
sleptons. Their symbols are the same as those of the SM ones, but they get an extra
tilde (˜). For example, the left- and right-handed selectrons are denoted by ẽL and ẽR .
The smuons and staus are given accordingly by: µ̃L , µ̃R , τ̃L , τ̃R . Only left-handed SM
neutrinos exist, the sneutrinos are therefore denoted by: ν̃e , ν̃µ , ν̃τ . Finally, the squarks
26
are given by q̃L , q̃R with q = u, d, s, c, b, t. Since the gauge interactions within the chiral
supermultiplet are equal, also the left handed squarks couple to the W boson whereas
the right handed squarks do not.
The Standard Model vector bosons reside in gauge supermultiplets. The gauge interactions of the SU (3)C group of QCD are mediated by the gluon g whose supersymmetric
partner is the gluino g̃. The spin-1 bosons W + , W 0 , W − and B 0 of the electroweak
gauge symmetry SU (2)L × U (1)Y have the winos and bino as supersymmetric partners:
W̃ + , W̃ 0 , W̃ − , B̃ 0 . As described in Sec. 1.2.4, electroweak symmetry breaking mixes
the W 0 and B 0 eigenstates to give the mass eigenstates Z 0 and γ. The corresponding
gaugino mixtures of W̃ 0 and B̃ 0 are the zino Z˜0 and photino γ̃.
The Standard Model Higgs boson (spin-0) resides in a chiral supermultiplet. It turns
out however, that one Higgs chiral supermultiplet is not enough. This would result in
what is called a gauge anomaly and would cause an inconsistent quantum theory1 . It
turns out that the problem can be solved by introducing two Higgs multiplets: one with
Y = + 21 and one with Y = − 21 2 . They are also both necessary as only a Y = + 12 Higgs
chiral supermultiplet can have the necessary Yukawa couplings to give masses to charge
+2/3 up-type quarks. The Y = − 21 Higgs chiral supermultiplet in its turn can only
provide masses for the -1/3 down-type quarks (and to the charged leptons). The scalar
fields with Y = + 12 and Y = − 12 are named Hu and Hd , respectively. The weak isospin
components of Hu with T3 = (1/2, −1/2) have electric charges 1,0 and are denoted by
(Hu+ ,Hu0 ). In a similar way we have (Hd0 ,Hd− ). The superpartners then become H̃u+ ,H̃u0
and H̃d0 ,H̃d− and are referred to as the Higgsinos.
The higgsinos and electroweak gauginos mix with eachother because of electroweak
symmetry breaking. The neutral higgsinos (H̃u0 , H̃d0 ) and the neutral gauginos (B̃, W̃ 0 )
combine to form four mass eigenstates called neutralinos (χ̃01 , χ̃02 , χ̃03 , χ̃04 ). The charged
higgsinos (H̃u+ , H̃d− ) and winos (W̃ + , W̃ − ) mix to form two mass eigenstates with charge
−
+
−
±1 called the charginos (χ̃+
1 , χ̃1 , χ̃2 , χ̃2 ).
The above discussed sparticles are all the particles comprised within the MSSM and
are summarised in Tables 2.1 and 2.2.
1
Gauge anomalies invalidate the gauge symmetry of a quantum field theory. Gauge symmetries
are theories where symmetry transformations are space-time dependent. They are used to generate
dynamics, namely the gauge interactions. For a more thorough explanation, refer to [27].
2
Y stands for weak hypercharge and is given by Y = 2(Q − T3 ), with Q the electric charge and T3
the third component of weak isospin.
27
Table 2.1: Chiral supermultiplets in the Minimal Supersymmetric Standard Model. The
spin-0 fields are complex scalars, and the spin-1/2 fields are left-handed two-component
Weyl fermions [26].
Table 2.2: Gauge supermultiplets in the Minimal Supersymmetric Standard Model.
The spin-1 fields are the SM vector bosons, the spin-1/2 ones are their supersymmetric
counterparts [26].
2.3.2
Broken Symmetry
In Sec. 2.2 it was explained how particles within the same supermultiplets must have
the same mass, this however requires some extra attention. At the time of writing,
not a single superpartner of the SM particles has been discovered. If supersymmetry
would be an unbroken symmetry, then there would have to be selectrons ẽL and ẽR with
masses exactly equal to the electron mass me = 0.511 MeV (this is also the case for all
other SM particles). Those sparticles should be easily detectable but they haven’t been
found. Supersymmetry must thus be a broken symmetry, causing sparticles to have
higher masses than their SM counterparts. In other words, this means that the model
explaining the symmetry breaking, should have a Lagrangian density that is invariant
under supersymmetry, but a vacuum state that is not.
28
The exact theoretical considerations fall outside the scope of this study, and it is satisfying enough to understand that superpartners will have higher masses than their SM
counterparts. The general idea however of supersymmetry breaking is to introduce an
effective MSSM Lagrangian consisting of two parts
L = LSUSY + Lsoft
(2.4)
with LSUSY containing the gauge and Yukawa interactions, preserving supersymmetry
invariance and Lsoft violating supersymmetry. The second part is called soft because
the breaking terms are constructed such, that they do not introduce quadratic divergences [28]. We also note that there are no theoretical predictions towards the actual
masses of SUSY particles, which makes SUSY searches extra challenging.
2.3.3
R-parity
R-parity RP [29], [30] is a discrete multiplicative symmetry that can be written as
RP = (−1)3B+L+2S ,
(2.5)
with B denoting the baryon number, L the lepton number and S the spin of a particle.
It has the property that all SM fields have RP = +1 whereas all the superpartners of SM
fields have RP = −1. R-parity is introduced into the MSSM in order to eliminate the
possibility of B- and L-number violation. It is mentioned here as R-parity conservation3
has three very important consequences:
• SUSY particles are always produced in pairs.
• The lightest sparticle with RP = −1, called the lightest supersymmetric particle
or LSP, must be stable4 . Considering it to be massive, weakly interacting and
electrically neutral, makes the LSP a viable candidate to explain dark matter.
It is clear that R-parity conservation puts some interesting constraints on the possible
SUSY Feynman-diagrams. Also note that the LSP behaves like a neutrino and thus
escapes detection, it is in general identified with the lightest neutralino χ̃01 .
3
R-parity conservation is an assumption made in the MSSM, theories with R-parity violation exist
too, but aren’t considered here.
4
Because of energy- and R-parity conservation it cannot decay to other SUSY or SM particles.
29
2.4
SUSY-searches at the LHC
Due to the fundamental differences between the SM and SUSY, it deserves our attention to understand how SUSY-searches at the LHC can be conducted. From theoretical
calculations, it seems that squark and gluino production are expected to be dominant
processes. Since they are expected to be heavy, they will produce long decay chains
eventually leading to a final state with only SM particles and LSPs. Those Standard
Model particles will typically be leptons (e or µ), neutrinos, and quarks that hadronise into jets. The LSP is neutral and interacts weakly, causing it to have the same
behaviour as the neutrino in a detector in turn causing it to be undetectable.
It becomes clear that one has to construct smart techniques in order for SUSY-signatures
to be detectable. Obviously, typical SUSY-processes will have large ETmiss due to the
presence of the LSP. This can be exploited by constructing various variables who can
discriminate SUSY- from regular SM-processes (an example will be given in the Analysis
section). SUSY searches are typically distinguished by the number of leptons required
in the final state. We list a few possibilities below.
• Zero-lepton searches typically focus on the escaping dark-matter candidate (the
LSP). The cross section for events with large ETmiss , large number of jets and no
leptons is very large. Unfortunately, the SM background shares this property,
making this a difficult channel to look at.
• One-lepton searches use the fact that SUSY models couple to weak interaction,
meaning it is likely that a single lepton (+ ν) is produced.
• A very SUSY-specific signal is the same-sign dilepton channel. A possible process
can be understood if we consider the decays g̃ → t̃t̄ and g̃ → t̃∗ t, which are possible
in simple R-parity violating models, and occur with the same probability. Gluino
pair-production could lead to events with two same-sign tops. If both decay
leptonically a same-sign dilepton signature is obtained. Obviously, opposite-sign
dilepton searches are also conducted, this will be more elaborately discussed in
Chapter 5
• Multi-lepton searches also exist in which production of charginos and neutralinos
are considered, giving rise to 3 leptons.
The above listing shows some of the possible SUSY-search scenarios, and more possibilities exist. This thesis will contain research work within an opposite-sign dilepton
analysis in which stop-quark production is the analysed signal in the MSSM, where
R-parity conservation is expected.
30
Chapter 3
The CMS Experiment at the LHC
3.1
The Large Hadron Collider
The Large Hadron Collider is the biggest and most powerful particle accelerator in
the world. It is located at CERN (European Organization for Nuclear Research) near
Geneva, Switzerland. The accelerator consists of a circular-shaped tunnel with a circumference of approximately 27 km and allows protons (as well as heavy ions) to
collide at very high center-of-mass energies.
In order to establish proton collisions, two beam pipes are necessary to bend the protons in opposite directions and allow collisions at fixed points along the accelerator.
The guidance and bending is done through the use of superconducting dipole magnets
to bend, and quadrupole magnets to focus the protons along the optical axis. Superconductivity is necessary as extremely high currents up to 12×103 A are required in
the magnet coils to provide high magnetic fields. Higher order magnets are also used
to provide additional corrections to the beam.
The design luminosity of the LHC is L = 1034 cm−2 s−1 , achieved from the circulation
of protons through the accelerator (about 11×103 turns per second) that are bunched
together with bunch spacings of 25 ns. Protons are
√ currently being accelerated to 6.5
TeV each, resulting in a center-of-mass energy of s = 13 TeV (before this, the LHC
also operated at 7 and 8 TeV at respective peak luminosities of 3.7 × 1033 cm−2 s−1 and
7.7 × 1033 cm−2 s−1 ). This is a never-before achieved energy and provides tremendously
exciting research opportunities.
On the LHC-ring (schematically depicted in Fig. 3.1), four main experiments are in31
stalled around the interaction points where the beams collide, namely, ATLAS and
CMS; two general purpose detectors, and ALICE and LHCb; conducting heavy-ion and
b-physics experiments, respectively. This thesis will be subjected to an analysis situated within the scope of the CMS-experiment, which will be more thoroughly explained
in the next section.
Figure 3.1: Schematic overview of the LHC-complex. Depicted in yellow are the four
c CERN Collaboration
main experiments. The injection of protons into the LHC is followed after a gradual increase in their energy
using different accelerator structures depicted in Fig. 3.1. The protons originate from
a bottle of hydrogen gas, whose electrons are stripped after applying an electric field.
They are passed to a Linear Accelerator (LINAC) structure, accelerating the protons
to an energy of 50 MeV before entering the booster. The booster accelerates to an
energy of 1.4 GeV, which subsequently passes the protons to the Proton Synchrotron
(PS) and eventually to the Super Proton Synchrotron (SPS), accelerating respectively
to 25 and 450 GeV. Finally, the protons are transferred to the two beam pipes of the
LHC, further increasing their energy up to 6.5 TeV at which they collide.
A schematic view of a proton-proton collision is given in Fig. 3.2. Two partons from
each of the two protons interact with each other forming a primary, hard-scatter interaction vertex (depicted with the red dot). This process produces a new state that
eventually decays into a spray of particles. Those undergo a showering process after
32
Figure 3.2: Schematic view of a proton-proton collision [31].
which they hadronise into the particles that finally reach the detector.
The huge advantage of a proton-proton accelerator over for example an electron-positron
accelerator is that a very wide energy range can be probed. This is a direct consequence
of the fact that partons inside the proton carry a fraction of the momentum of the proton, thus allowing collisions to take place at different values of transferred momentum.
With proton-proton collisions, the LHC allows to look at the fundamentals of what matter consists of and helps scientists to gain a deeper and more profound understanding
of the fundamental open questions in physics. This concerns the basic laws governing
the interactions among elementary particles, the scales at which current theories break
down or not, etc.
Apart from proton-proton collisions, the LHC also serves another purpose: heavy-ion
collisions. Reactions between heavy-ions have proven themselves very useful regarding
the investigation of the thermodynamical properties of QCD matter. The difference
with protons colliding is not that single protons, but a bunch of protons and neutrons
are smashed together to form a very dense nuclear matter system. With these type of
collisions, physicists try to investigate the quark-gluon plasma.
33
The quark-gluon plasma is a hypothesized phase of nuclear matter, believed to exist at
either very high densities or very high temperatures (or both). If one were to compress
a nucleus (without altering the temperature) to very high densities (∼10×), individual
nucleons would start overlapping. If one on the other hand would increase the temperature (without altering the density), the individual nucleon-nucleon interactions would
increase in such a way, that the collisions between them would make it impossible to
assign a quark or gluon to any hadron in particular. The state is said to consist of
asymptotically free quarks and gluons.
The discovery of the quark-gluon plasma could provide great confidence in our understanding of strongly interacting matter. Apart from that, it would be a simulation of
the universe at a very early stage in its history (∼30 µs after the Big Bang).
At CERN, the LHC operates approximately one month yearly in nuclear collision mode.
Pb-nuclei collide at 2.76 TeV per nucleon pair and produce big sprays of strongly interacting particles around the central collision point who are then detected by the detectors
installed on the LHC.
Figure 3.3: Two heavy-ions collide with partial overlap (left). A spray of strongly
interacting particles (black dots) are produced (right) [1].
34
3.2
The CMS-Experiment
The CMS, or Compact Muon Solenoid experiment, is a general purpose particle physics
detector situated on the LHC-ring. It is cylindrically shaped with a diameter of 15 m
and a length of 28.7 m, weighing 14000 metric tonnes. It gained world fame after it
discovered together with ATLAS the Standard Model Higgs-Boson in July 2012. The
main objectives of the CMS-experiment are (1) to explore physics at very high energy
scales (TeV-range), (2) to test the SM at new energy regimes and phase space regions
that have not been explored before, (3) to increase the precision of SM electroweak and
QCD parameters, (4) to study the properties of the recently discovered Higgs-boson,
(5) to look for physics beyond the standard model, in particular Supersymmetry and
(6) to help in providing a possible explanation of dark matter.
The detector has an onion-like structure where each layer serves a different purpose that
helps in identification of particles originating from the central collision. The signal from
each detector component is then combined in order to reconstruct the original process
that took place. This is not an easy task as collisions are happening at a frequency of 40
MHz, making extremely fast electronics and smart algorithms necessary to distinguish
the different interactions. The different components of the detector are described below.
c CMS Collaboration
Figure 3.4: Overview of the different sections of CMS. 35
3.2.1
The Tracker
The tracker sits at the very core of the detector and surrounds the interaction point.
It has a length of 5.8 m and a diameter of 2.5 m and covers a pseudorapidity range of
|η| < 2.51 . The tracker’s function is to provide a precise measurement of the trajectories
of charged particles emerging from the interaction point as well as identifying secondary
vertices from prolonged b quark and τ -decays. Due to the extremely intense dose of
radiation at which the tracker is exposed, it is composed entirely from silicon-based
detector technology, as this material provides a very long lifetime.
The tracker consists of two main sections. The most central part is the pixel detector
consisting of three co-axial cylindrical layers of pixel detector modules, closed of at the
side by another two pixel modules. The pixel detector delivers three high-precision
space points for each particle trajectory (charged particle trajectories are curved due
to the presence of a high magnetic field, to be explained later). It is this precision that
allows accurate measurements of secondary vertices as displayed in Fig. 3.5.
In order to detect b quarks (typically necessary for detecting processes as depicted in
Fig. 4.3.1) one determines the impact parameters d (the distance of closest approach
of a secondary track to the primary vertex) and determines whether they’re larger than
some reference to be sure the track originated from a secondary vertex. A distribution of
Lxy (the decay distance in the plane transverse to the beam direction) allows comparison
with typical b quark lifetime distributions.
Figure 3.5: Display of a secondary vertex originating from b-quark decay. d and Lxy
c CDF Collaboration
are also depicted. The second and outermost section of the track system is the silicon strip tracker and
1
The pseudorapidity
η is a measure for the angle of a particle relative to the beam-axis and is
defined as η = −ln tan θ2 . It is frequently used as η is boost-invariant, whereas θ isn’t. This allows
displays of particle energy depositions in so-called lego plots between the two boost invariant quantities
η and φ, with the latter being the (boost-invariant) angle in the transverse plane. This way, angular
differences are not influenced by the boost of the particles.
36
provides additional resolution to the tracks measured. After the particles leave the pixel
detector, they pass through 10 layers of silicon strip detectors, reaching out to a radius
of about 1.2 m. Silicon semiconductor technology allows very fast readout capability,
which is necessary for collisions happening each 25 ns.
3.2.2
The Electromagnetic Calorimeter
The electromagnetic calorimeter or ECAL surrounds the tracker, with an inner radius
of about 1.3 m to an outer radius of about 1.7 m and covers a pseudorapidity range
|η| < 3.0. Its function is to detect those particles that interact electromagnetically;
in particular electrons/positrons and photons. It consists of a large collection of lead
tungstate crystals (PbWO4 ) spread over the barrel and two endcaps. These kind of
crystals scintillate, meaning they produce light proportional to the energy deposited in
the crystal. This light is collected through Avalanche Photo Diodes (APDs) attached
to one side of the crystal2 which eventually produce an electric signal in proportion to
the incoming light, thus in proportion to the deposited energy. The crystals have a very
short interaction length and can withstand very high doses of radiation.
PbWO4 is an inorganic scintillator crystal with impurities added. When an energetic
particle passes, electrons are elevated from valence to the conduction band. The electron is then free to move through the crystal after which it eventually recombines with
a hole and produces light. The impurities are necessary to make the recombination
more efficient, thus yielding very fast light outputs (See Fig. 3.6). In 25 ns, about 80%
of the light produced from the energy deposit is emitted [32].
c G. F. Knoll, Radiation
Figure 3.6: Schematic display of the PbWO4 band structure. detection and measurement. (2010)
2
APDs are used for the ECAL barrel section. For the ECAL endcap sections, vacuum phototriodes
are used.
37
At the endcap on the inner side, the ECAL preshower system is installed on both sides.
It has a finer granularity; silicon detector strips of 2 mm wide compared to the 3 cm
crystals of the rest of the ECAL. One of its important design constraints is to be able to
resolve nearby γ’s so that the Higgs boson decaying into two photons could be precisely
reconstructed. This is studied using the π 0 → γγ process which is typically boosted and
produces very small decay angles between the 2 photons, thus giving rise to a possible
single hit in the 3 cm ECAL crystal. The smaller strips can distinguish between such
hits.
3.2.3
The Hadronic Calorimeter
The hadronic calorimeter or HCAL is situated in between the ECAL and the solenoidal
magnet (next section) with inner radius of 1.77 m and outer radius of 2.95 m and covers a pseudorapidity range |η| < 3.0. It is designed to measure mainly the energy of
hadrons, interacting strongly with the HCAL material. Due to the larger interaction
length of hadrons, the HCAL must be larger than the ECAL.
The HCAL is a sampling calorimeter3 , existing of 36 barrel and endcap wedges. It is
made of alternating layers of dense brass (passive) absorber material and (active) plastic
scintillator. The idea is that particles hit the absorber material and produce a cascade
of shower particles. These deposit energy in the scintillation material again providing
an electrical output proportional to the deposited energy. This is done through the use
of optical fibres that conduct the light output into readout boxes where photodetectors
amplify the signal. The shower has an electromagnetic and hadronic component (see
Fig. 3.7). The hadronic component contains mostly π ± and π 0 that decay according to
π ± → µ± + ν and π 0 → γγ. The hadronic and electromagnetic components of showers
fluctuate grossly event by event.
The HCAL thus provides information on the position of a particle, energy and arrival
time. It is made as hermetic as possible, which allows indirect measurements of noninteracting particles. This is thanks to the fact that the sum of all transverse momenta
should equal one, which follows from momentum conservation and the knowledge that
pinitial
= 0. The missing transverse momentum is then defined as pmiss
= -Σi piT , where
T
T
the sum i runs over all track momenta.
3
In a sampling calorimeter, the absorber material degrades the energy of the particle hitting the
detector and this interaction produces a shower of particles. The absorber is alternated by layers of
active material that produces the signals that could be read out.
38
Figure 3.7: A neutron hitting the absorber material and starting a particle shower, with
both an electromagnetic and hadronic component. [33]
3.2.4
The Superconducting Solenoidal Magnet
The superconducting magnet for CMS surrounds the tracker, the ECAL and the HCAL.
It was designed to reach a 4-T magnetic field originating from a unique 4-layer solenoidal
winding of NbTi conductor wires. The magnet is 6.4 m in diameter and 12.5 m in length,
it weighs 220 metric tonnes. In order to reach the large magnetic field, very high currents must flow through the solenoidal wires. This is only possible if those wires are
superconducting, which needs cooling of the magnet towards cryogenic temperatures of
4.5 K, done using liquid He.
A 4-T magnetic field of this size is truly remarkable. Particles with very high momentum originating from the interaction point are only bent in strong magnetic fields
and this allows CMS to pinpoint a particle’s momentum with high precision. Fig. 3.8
shows how different particles are bent under influence of the magnetic field, as well as
the location of detection.
The solenoid is surrounded by a 12.5 t iron return yoke (displayed in red in Fig 3.4
and 3.8), which contains and guides the magnetic field. It also functions as the main
support of the entire detector.
39
Figure 3.8: Schematic view of CMS along the beam axis. Different particles movement
and interaction locations are displayed.
3.2.5
The Muon System
The detection of muons is of particular importance for CMS, as is clear from its name.
CMS is designed in such a way that muon detection can be done with very high precision.
Muons are not stopped by CMS’s calorimeters. The muon system is installed at the
outer edge of the detector, where they are likely to be the only particles left to register a
signal. A particle is measured by fitting a trajectory to a track from hits created in the
four muon stations, which are situated outside the magnet coil and have return yoke
plates in between them (denoted in red in Fig. 3.8). This combined with the tracker
hits at the core of the detector allows a particle’s path to be precisely reconstructed,
especially thanks to the high magnetic field to bend even the paths of very high-energy
muons.
Hits in the muon chambers originate from a combination of three types of gaseous
particle detectors:
(1) Muon drift tubes: These are rectangular shaped gaseous volumes (Ar/CO2 )
whose edges consist of cathode strips and through which an anode wire runs.
A charged particle passing through the volume generates ionisation around the
particle track where positive ions and negative electrons respectively move to the
40
cathode and the anode. Stacking groups of drift tubes on top of each other, allows
from timing measurements of charge collection time to reconstruct the path of the
muon.
(2) Cathode strip chambers: These components are situated at the endcap disks
of the detector. They consist of positively-charged anode wires who are perpendicularly crossed with negatively-charged cathode strips, all confined within a gas
volume (Ar/CO2 /CF4 ). Due to the particular setup of those chambers and the
electrical fields applied, electrons generated from the ionisation process trigger an
avalanche-effect allowing precise measurements of the particle’s position.
(3) Resistive plate chambers: These components are installed both on the barrel
and endcap sections of CMS. They consist of two parallel plates, a positively
charged anode and a negatively charged cathode, both made of high resistivity
material, separated by a gas volume (C2 H2 F4 /i-C2 H10 ). They are also operated
in avalanche mode and provide very fast detector timing.
3.2.6
Triggering and Data Acquisition
When bunch crossings take place each 25 ns, about 1 million pp collisions take place
every second4 . It is technically impossible to detect and read out all the data from
each individual collision. Apart from technical reasons, plenty of those collisions are
worthless as they might be low-energy collisions for instance, instead of highly energetic, and would be rejected from all interesting analyses in event selection. One clearly
needs techniques to only select those events that are potentially interesting. Due to the
tremendously high frequency of collisions, new events are happening, when the previous one did not even leave the detector yet. This is called pile-up and complicates the
selection of interesting events, degrades measurements and sensitivity to new physics
searches if analyses are not accoumpanied with sophisticated pile-up mitigation techniques.
Everything that gets measured by CMS needs to pass a set of initial criteria before
further, more strict selection criteria are imposed on events in the high level trigger and
in specific analyses. This process is called triggering. The triggering happens in two
stages in CMS. The first stage is the Level 1 (L1) trigger [34] and the second stage is
the High Level Trigger (HLT) [35], [36]. The L1 trigger makes use of the calorimetry
and muon systems. Its decision is based on the presence of particle candidates such
as electrons, muons, photons and jets above certain ET or pT tresholds. It effectively
4
On average about 20-30 protons collide each bunch crossing.
41
reduces the 40 MHz event rate to 100 kHz. The HLT filters the rate even further to
100 Hz. The CMS DAQ/HLT system has the following functionality
• Read-out after a Level-1 trigger accept.
• Execute physics selection algorithms on the events that are read out. Accept
those events that are physically most interesting.
• Forward these events towards mass storage for offline computing.
The HLT selection algorithms [36] use info from calorimeters, muon systems and pixel
detectors. Then, the algorithms do primary vertex and full track reconstruction and btagging using secondary vertex identification (Fig. 3.5). It needs to be noted however,
that to minimize the CPU required by the HLT, the algorithms only reconstruct the
information partially. Full event reconstruction is done in offline computing.
42
Chapter 4
Modelling and Analysis Software
This chapter will provide an overview of the software used in this thesis to perform
the necessary physics analyses. A general outline of the CMS Software (CMSSW)
Application Framework will be given as well as an introduction to Monte Carlo (MC)
event generators. An overview of the different data-types that are relevant may be
beneficial:
• RAW: Detector data after it passed the L1 trigger and HLT selection criteria
(see Sec. 3.2.6).
• RECO: Reconstructed objects (tracks, vertices, jets, leptons, etc.) from all stages
of reconstruction.
• AOD: Subset of RECO data that is relevant for physics research in a convenient
and compact format.
• GEN: Generated Monte Carlo (MC) data.
• SIM: Energy depositions of MC particles in the detector.
• DIGI: SIM hits converted into digitised detector response. This is the equivalent
of RAW data, but here with simulated MC data.
4.1
CMSSW Application Framework
CMSSW [37] is a framework that provides the ability to deploy reconstruction and
analysis software in order for physicists to be able to perform an analysis. It is built
around an Event Data Model (EDM). An event in this context denotes a C++ object of
43
all RAW and reconstructed data for one particular collision. All the objects in such an
event, are stored in a particular format called ROOT files. ROOT is an object-oriented
framework aimed at solving the data analysis challenges of high-energy physics [38].
An important software toolkit used in CMSSW is GEANT4 (GEometry ANd Tracking)
[39]. It is a platform for simulation of passage of particles through matter. It is used
in a large variety of high-energy physics experiments. For CMS in particular, it allows
MC data to be reconstructed as if it were real data. This means effects taking into
account the geometry of the detector that can influence the reconstructed events are
included.
As was mentioned in Sec. 3.2.6, the 40 MHz event rate is reduced to about 100 Hz of
RAW data events. This data is then processed towards RECO data objects. The latter
dataset is still very large and contains irrelevant information for physics analyses and
gets reduced to AOD data (Analysis Object Data).
4.2
Particle-flow Event Reconstruction
Particle-Flow (PF) [40, 41] is an algorithm to reconstruct all stable particles in an event
by combining the kinematical information from the different CMS detector parts. Photons deposit their energy in the ECAL and are thus reconstructed from ECAL clusters.
The energy of electrons is determined from a combination of the electron momentum
acquired by the tracker, the energy deposited in the ECAL cluster and the total energy
originating from Bremsstrahlung photons that are compatible with the electron track.
The energy of muons is determined from curvature measurements of its track, obtained
from tracker and muon system information. The energy of charged hadrons is measured from a combination of tracker and ECAL and HCAL cluster information, when
both values are found to be compatible1 And finally, the energy of neutral hadrons is
obtained from ECAL and HCAL clusters.
4.3
Monte Carlo Event Generators
MC generators are indispensable in hadron collider physics research. They comprise of
a set of models to simulate actual events from a particular process. For the LHC, when
1
If the calorimeter response is to be too small, a cleaning procedure is performed to remove potential
spurious or mis-reconstructed tracks. If, on the other hand, the calorimeter response is too large, the
PF algorithm assigns the energy excess to a photon and possibly a neutral hadron.
44
two protons collide, a few very important things must be taken into account in order
to produce a viable simulation of the interaction event. First of all, a hard scattering
process must be implemented. This is the interaction between two partons (quarks or
gluons) from each of the two protons. The hard scattering occurs at high momentum
transfer and is described by the matrix element (ME). Furthermore, it can be computed in the perturbative regime thanks to the running of αS (Sec. 1.2.3). Afterwards,
the process of parton showering (PS) will start. This means that quarks and gluons originating from the hard scattering are propagated and radiate additional quarks
and gluons and finally undergo hadronization. Here, a transition from perturbative
to non-perturbative regime occurs since every additional radiation lowers the amount
of transferred momentum. The sequential steps in this process can be modelled using
dedicated Monte Carlo programs. This means the modelling is done in a probabilistic
way, which is an accurate description of what happens in nature. An introduction on
two concrete and distinct examples of MC programs, POWHEG and PYTHIA, will be given
in order to gain a better understanding of their functionality and use.
4.3.1
Powheg
POWHEG [42], [43] is a Matrix Element (ME) Monte Carlo event generator. Given a particular process, e.g. pp → tt̄, it will generate all the possible Feynman diagrams that
can give rise to the tt̄ state at a particular order in perturbation theory2 . It must be
noted that this type of MC generators work in the perturbative regime. As was mentioned in Sec. 1.2.3, the strong coupling constant αS is a running constant, meaning
it becomes smaller the larger the amount of transferred momentum gets. As POWHEG
simulates the collision of quarks or gluons from a pp collision, there is no momentum
lost to radiative processes yet, so the value of transferred momentum is very high. A
scattering event in this regime is called a hard-scattering.
POWHEG is able to calculate matrix elements up to Next-to-Leading-Order (NLO). This
means that the generator also takes hard parton emission into account. Examples of
LO and NLO diagrams are given in Figs. 4.1 and 4.2. They display examples of tt̄production diagrams, the f and f¯ are a fermion and an anti-fermion3 . Although the
diagrams seem simple, NLO diagrams provide difficulties in theoretical calculations.
2
3
To view which processes can be computed with POWHEG, see [44].
The W-boson has multiple ways of decaying, e.g. W → lν or W → q q¯0
45
Figure 4.1: First order (LO) tt̄ production Figure 4.2: Example of a tt̄ production proprocess without additional radiations.
cess with a gluon radiating from one of the
initial quarks. The gluon gives rise to an
extra vertex, causing the diagram to be at
NLO.
NLO diagrams provide not only positive, but also negative contributions to the crosssection. This makes it difficult to implement full NLO processes in MC generators.
Another difficulty exists when the ME generator (e.g. POWHEG) is made to match with
a PS code (e.g. PYTHIA). This is called merging, and will be explained in Sec. 4.3.2.
4.3.2
Pythia
PYTHIA [45] is used to generate high-energy-physics events: sets of outgoing particles
produced from the interactions between two incoming particles. It comprises a collection of physics models for the evolution of a hard scattering process to a complex
multihadronical final state4 . The complexity stems also from the fact that many processes must be taken into account.
First of all, the original Feynman-diagrams receive Strahlung-modifications, in which e
→ γe or q → gq type of processes result in additional final-state particles. Second, there
are higher-order Feynman diagrams which sometimes combine to cancel divergences,
resulting in perturbative techniques to become necessary. Typically this is only done up
to one loop, which is referred to as an NLO-approximation. Third, due to confinement
of quarks and gluons, no coloured particles can propagate freely. Therefore they must
hadronize in colourless final states. This finally results in a collection of jets composed
of photons, hadrons and leptons.
4
Note that PYTHIA usually does not simulate the interaction itself but is typically used for the PS.
46
Figure 4.3: An example of a PS process calculated in PYTHIA. Starting of with a transferred momentum of q12 , the parton radiates gluons losing momentum to q22 , q32 , etc.
Once the parton lost enough momentum and passed a certain cut-off momentum, it is
left to hadronise into a colourless final state.
The entire problem can be summarised in two steps: first, the hard process is used as
input to generate the corrections described in the previous paragraph and second it is
left to hadronize. It is the task of PYTHIA to generate events (using MC techniques)
such that its output (after simulation of the detector using GEANT4) is very similar to
real data as measured by CMS. In typical data analysis, this is a very important aspect
of the conducted study, since such generated events provide a tool for devising analysis
strategies that can be used on real data. Furthermore they are of essential importance
as they allow to model SM processes, make SM parameter predictions, and model backgrounds.
It was alluded in Sec. 4.3.1 that difficulties exist when the ME generator functions
at NLO. Since PYTHIA indeed produces a PS it is possible that a PS produced from a
LO diagram, coincides with a diagram produced in NLO (for an example of a PS, see
Fig. 4.3). The concept of merging [46] solves this by including algorithms in the MC
generators to prevent double counting, so that the same diagrams aren’t added to the
phase space5 .
4.3.3
Tuning of event-generators: Professor
Monte-Carlo event-generators use various approximations. To ensure the validity of
those approximations, tuning is done on its parameters, to find a match between the
predicted set of events compared to what’s measured in data. With Eq. 1.8, we alluded that there was a perturbative and non-perturbative part to be taken into account
to compute the cross section. Both the perturbative and the non-perturbative parts
5
The phase space is the mathematical space of all possible momenta of all the outgoing particles.
47
could be tuned. The UE has many parameters that can not be calculated theoretically,
therefore tuning becomes necessary to have these parameters implemented correctly.
PS programs also have some tunable, free parameters such as αSISR . There’s multiple
ways for doing so: either one does it manually which requires great skill and luck, or
one does it using dedicated programs optimised for tuning MC-generators in the most
efficient way.
Professor [47] is a program dedicated to tune model parameters of MC event-generators
to experimental data by parametrising the per-bin generator response to parameter
variations and numerically optimising the parametrised behaviour. Typical eventgenerators have O(10-30) parameters that are relevant for collider physics simulations.
It goes without saying that sampling in a multi-dimensional hyperspace is computationally very costly. If P is the number of parameters to be tuned, then for each variation
of the P -element parameter vector p = (p1 , ..., pP ) a new event-generator computation
would be necessary. The idea here, would be to define a goodness of fit (GoF) function
between a limited set of MC data and the actual real data, and then to minimise that
function corresponding to an optimal parameter vector pbest . A more elaborate discussion on the functioning of Professor can be found in appendix A.
48
Chapter 5
Analysis Part I: Search for Scalar
Top Quark Partners
5.1
Introduction
The first part of the analyses done in this thesis revolves around a SUSY search for
∗
stop
√quark (t̃t̃ ) production conducted in the opposite sign dilepton channel at the LHC
at s = 13 TeV pp collisions. We look for the signal in which both W bosons decay
leptonically (i.e. to ee, µµ or eµ), hence the name: dilepton channel. It is depicted in
Fig. 5.1.
q̄
b
t
W
t̃
g
t̃
+
l+
νl
χ̃01
χ̃01
∗
W−
t̄
l−
ν̄l
b̄
q
Figure 5.1: A hypothetical t̃t̃∗ pair is produced after a pp collision at the LHC. When
assuming R-parity conservation, the stop quarks are produced in pairs and decay to a
top quark and a neutralino (the latter is the LSP, invisible in our detector). The W
bosons, originating from top quarks, both decay leptonically.
49
Figure 5.2: Branching fractions of top quark decay. Each time the decays of the two
W bosons are denoted. Full hadronic means both the W-bosons decay into two quarks
and thus produce jets. Note that the decay of the unstable τ -lepton as τ → e/µ + νe/µ
has not been included.
The branching fractions of the top quark decay are given in Fig. 5.2. It is clear that
the dilepton channel accounts for only a small fraction of top pair decays (9%). The
full hadronic channel for example, accounts for a branching of 46%. There are plenty
of processes that have large jet activity making it difficult to discriminate signal from
background. The dilepton channel on the other hand produces electrons or muons1 .
Those are very easily detectable with the CMS ECAL (Sec. 3.2.2) and Muon systems
(Sec. 3.2.5).
In SUSY-searches, there is another very important difference compared to regular SMsearches. There are no a priori predictions on the masses of the supersymmetric particles. This means that the energy available for the top quarks depends on the masses
of the stop t̃ and neutralino χ̃01 , which aren’t known. One therefore always has to make
a prediction on the mass difference ∆(mt̃ , mχ̃01 ). Depending on this mass difference,
more or less energy will be available for top quark production and this will have a
consequence on how energetic its decay products are.
A particularly difficult signal region to search for is the compressed signal, for which
∆(mt̃ , mχ̃01 ) is small. This means the energy available to the top quark is only 100
1
τ
+
The tau-lepton dominantly decays into electrons or muons through τ − → ντ + e− /µ− + ν̄e /ν̄µ and
→ ν̄τ + e+ /µ+ + νe /νµ .
50
GeV/c2 compared to the 173.3 GeV/c2 needed to produce it on-shell. The top quark
will therefore be off-shell and produces very soft b quarks, leptons and neutrinos.
Due to the presence of two LSPs and two neutrinos from W decay, the signal has a large
amount of ETmiss , which will typically contain more ETmiss than regular SM processes
which can be exploited by constructing a particular type of variable, MT 2 (ll). This
variable will provide a way of making a distinction between the signal region and the
background region.
5.2
MT 2(ll)
We now introduce an important variable for this study called MT 2 (ll) [48] offering the
possibility of selecting a signal region. The variable is defined by
MT2 2 (ll) =
min
miss
miss
pmiss
l1 +pl2 =ET
miss
2
l2
miss
max MT2 pl1
,
p
,
M
p
,
p
T
l1
T
T
l2
(5.1)
miss
= ETmiss means
and is constructed as listed below. Note that the condition pmiss
l1 + pl2
miss
miss
miss
is defined in Sec.
= |pmiss
that |pmiss
T |. The variable pT
l1 | + |pl2 | must equal ET
3.2.3.
and pmiss
(1) The ETmiss is partitioned into two hypothetical neutrinos with pmiss
l2 .
l1
(2) The transverse mass MT is computed for each lepton with one of the hypothetical
neutrinos.
(3) The maximum of those two values is stored, the other is discarded.
(4) This is repeated for every partition of ETmiss and then the minimum is taken. That
value is the event’s MT 2 (ll).
Note that if in the computation for MT 2 (ll) a maximum of 0 GeV/c2 is found for a
given partition of ETmiss , MT 2 (ll) = 0 GeV/c2 will be kept. For a particular set of events
this will always be the case when the ETmiss lies within the smaller angle subtended by
the pT of the two leptons. For those events, a partition will be encountered with the
two neutrinos aligned with the pT of the leptons causing to yield a value of MT = 0
GeV/c2 . Therefore a peak in the MT 2 (ll) spectrum at 0 GeV/c2 will be typically seen.
For dileptonic tt̄ and W W events (some of our SM backgrounds), it can be shown [49]
that MT 2 (ll) has the same property for leptonic and single W events, namely that the
51
distribution has an endpoint at the mass of the W boson. However, two neutralinos as
illustrated in the process in Fig. 5.1 give rise to additional ETmiss . This will cause the
partitioning of ETmiss in Eq. 5.1 to create more energetic neutrinos than they actually
are, thus causing MT 2 (ll) to possibly have a higher value than for example a regular tt̄
process.
For top squark pair events, the distribution in MT 2 (ll) will depend on the value of
∆(mt̃ , mχ̃01 ). For large top squark masses, the neutralinos are boosted and this additional ETmiss will push the MT 2 (ll) distribution past the value of MW . If the mass
difference between the top squark and top quark is small, the neutralinos will be produced relatively at rest and will cause the MT 2 (ll) distribution to look like the one
found from SM tt̄ events.
5.3
Analysis strategy
This section will provide an overview of the analysis strategy taken to construct an optimal selection2 . The selection requires plepton
> 20 GeV/c, |η| < 2.4, miniRelIso < 0.2
T
(see Sec. 5.5), dxy < 0.05, dz < 0.1 for the leptons, and pjet
T > 30 GeV/c, |η| < 2.4 for
the jets. An overview of the Feynman diagrams of the most important MC generated
backgrounds (processes that give rise to the same signature as the signal) is given on
page 563 .
It is easily seen from the Feynman diagram in Fig. 5.1 that we will have to put requirements on Njets and Nb-jets . Therefore, at least two jets are required of which at least
one is b-tagged.
The most dominant background consists of Drell-Yan (DY) production: two leptons
with opposite signs originate from Z boson decay. It needs to be drastically reduced
by applying specific cuts. A regular DY process does not have any ETmiss . It is possible
however, that a gluon radiates from the initial incoming quark pair. This produces a
jet opposite to the lepton pair, as depicted in Fig. 5.3. If the pT of this jet is over- or
undermeasured in the detector, fake ETmiss is reconstructed.
2
The selection should leave the number of signal events as untouched as possible, but should reduce
backgrounds as much as possible.
3
The backgrounds from QCD- and tt̄ + X production (with X a W or a H boson) are not given
there but are also present in the analysis. They are very similar to process (3), but with a W - or
H-boson radiated from the top quark instead of a Z 0 boson.
52
Since we require at least two jets, it is possible that two gluons are radiated from the
incoming quark pair, and thus their energies get overmeasured4 . Depending on overor undermeasurements, the ETmiss is opposite to the jet or aligned with the jet. By
requiring cos(φETmiss − φjet1,2 ) < cos(0.25), events with jets aligned and opposite to the
ETmiss will be omitted from the analysis. A second pair of cuts that effectively reduces
DY is cutting directly on ETmiss and the ETmiss -significance (see Ref. [50]). The latter is
defined by
E miss
ETmiss -significance = √T
HT
(5.2)
√
with HT equal to the sum of the pT of all reconstructed jets. HT is a measure for the
resolution of the reconstructed ETmiss . The chance that invisible particles contributed to
ETmiss is significantly higher if the value of ETmiss is larger than its resolution. Requiring
the event to have an ETmiss -significance larger than a certain treshold reduces the DY
background as the ETmiss will be of the order of the resolution. This won’t affect our
signal as it should have a large ETmiss due to the neutralinos.
Figure 5.3: A jet radiating in the opposite direction of the lepton pair can get over- or
undermeasured and account for fake ETmiss . If the jet energy is overmeasured, we have
∆(φETmiss , φjet ) ≈ π. If it is undermeasured, we have ∆(φETmiss , φjet ) ≈ 0
4
Jet mismeasurements occur from noise effects, varying detector response, pile-up contributions,
etc.
53
To veto events in which the leptons originate from Z decay, a Z-veto cut is applied.
Concretely, we require |mll − mZ | > 15 GeV/c2 for same flavour (SF) leptons with mll
equal to the invariant mass constructed from both leptons. As the prompt leptons coming from W decay are highly energetic, a reconstructed invariant mass cut of mll > 20
GeV/c2 is required.
All the applied cuts (including some that weren’t mentioned, but are trivial) and their
effect on the backgrounds and signal are given in Table 5.1 in a cutflow diagram. Notice
how all backgrounds are efficiently reduced by the MT 2 (ll)-cut except for tt̄ + jets, tt̄ +
Z and tt̄ + X. The latter two are easily explained as they are irreducible backgrounds.
This can be understood by looking at the tt̄ + Z diagram (5) on page 56. The diagrams
for tt̄ + X are not given, but contain tt̄ + W and tt̄ + H production. If the Z boson
or H boson decay gives rise to two neutrinos, then these backgrounds mimic our signal
precisely causing them to end up in the high MT 2 (ll)-tails. The presence of the tt̄ +
jets background is not easily explained and requires more attention (Sec. 5.4).
54
Cut
Two well reconstructed leptons
Opposite Sign (e± e∓ , µ± µ∓ , e± µ∓ )
mll > 20 GeV/c2
|mll − mZ | > 15 GeV/c2 for SF
≥ 2 jets
≥ 1 b-tags
ETmiss > 80 GeV
√
ETmiss / HT > 5
cos(φETmiss − φjet1,2 ) < cos(0.25)
MT 2 (ll) > 140 GeV/c2
Cut
Two well reconstructed leptons
Opposite Sign (e± e∓ , µ± µ∓ , e± µ∓ )
mll > 20 GeV/c2
|mll − mZ | > 15 GeV/c2 for SF
≥ 2 jets
≥ 1 b-tags
ETmiss > 80 GeV
√
ETmiss / HT > 5
cos(φETmiss − φjet1,2 ) < cos(0.25)
MT 2 (ll) > 140 GeV/c2
DY
13333791.3
13281921.7
13189194.4
1185896.0
52475.3
5048.1
84.2
61.0
46.9
0.4
W + jets
5554.2
3679.0
3357.3
3140.3
928.4
86.7
16.8
14.9
13.3
0.1
tt̄ + jets
218650.8
214128.7
208590.9
185391.0
137452.0
108792.2
44724.8
41028.4
38318.5
2.9
Diboson
39734.0
38301.2
37437.8
18819.1
2475.6
206.7
54.7
50.4
47.8
0.3
tt̄ + Z
496.2
450.0
439.6
222.1
204.8
155.2
76.9
63.3
58.2
1.6
Triboson
142.7
132.1
131.1
48.1
28.4
5.6
2.7
2.4
2.3
0.0
tt̄ + X
884.5
687.6
673.3
436.5
399.7
315.3
163.6
130.1
119.1
1.1
QCD
255025.4
233498.5
37467.9
36338.5
1860.0
67.3
0.0
0.0
0.0
0.0
Single t
20550.0
19900.4
19413.5
17378.8
6497.2
4567.2
1940.3
1814.0
1699.0
0.6
Signal
296.8
293.5
291.1
261.1
206.0
168.7
149.8
146.3
136.4
26.1
Table 5.1: Cutflow-diagram of the applied cuts in the selection for the OS dilepton
t̃t̃∗ -search. The numbers represent the number of MC events produced. The signal is
found on the bottom right, the precise value of ∆(mt̃ , mχ̃01 ) is not given here as it is of
c Schöfbeck R.
less importance. 55
(1) Drell-Yan production
(2) tt̄ + jets production
q̄
b
t
g
t̄
l+
W+
νl
W−
l−
ν̄l
b̄
q
(3) Single top production
(4) W + jets production
q0
b
g
b̄
l
W
ν
q̄
(5) tt̄ + Z production
(6) Diboson production
q0
t
νl
W
Z0
t
+
ν̄l
g
l+
l−
νl
t̄
W
q̄
−
ν̄l
(7) Triboson production
q
0
νl
W+
W+
l+
q̄
Z0
q
W
q̄
−
l
1
−
ν̄l
56
5.4
MT 2(ll)-tail events study for the tt̄ + jets background
The large amount of tt̄ + jets events in the tails of MT 2 (ll) is worrying. This is due to
the fact that following from Eq. 5.1 no tt̄ events should end up in the high tails since
the only ETmiss can originate from the two neutrinos. We define the high-tail region
as MT 2 (ll) > 140 GeV/c2 . For this region an explicit check will be done to find out
what the kinematics of tt̄ + jets events exactly look like in full detail. The tt̄ + jets
background in function of MT 2 (ll) is displayed in Fig. 5.4.
Figure 5.4: Number of tt̄ + jets events in function of MT 2 (ll). The expected cutoff around 80 GeV/c2 is visible (notice the log scale), however there are still events
surviving into the high-tail region (> 140 GeV/c2 ).
A huge advantage in this study, is that in MC we have access to the generated particle’s kinematical information (GEN info), which is not the case in data. We can thus
explicitly see if something went wrong in the process of reconstructing the particles. In
reconstruction (RECO), the interactions in the detector are also accounted for, which
inevitably causes incorrect reconstructions of particles. We are particularly interested in
the leptons that were selected as the prompt leptons to see whether they are in fact the
decay products of the W bosons. If not, we are selecting the wrong leptons to compute
the event’s MT 2 (ll) and that could explain why we see events ending up in the high tails.
57
GEN - Level
RECO - Level
Access to simulated
kinematical information.
→ Monte Carlo
⇐⇒
Access to reconstructed
kinematical information.
→ Monte Carlo
→ Data
In the tail-region (MT 2 (ll) > 140 GeV/c2 ), 37 tt̄ + jets events are counted. For those
events, the GEN-information is matched to the RECO-information, of which an example
is displayed in Table 5.2. This means that for each of the two RECO leptons, the
corresponding GEN lepton is identified. For the GEN leptons, the PdgIds5 of the
mother particles are checked (the one from which the lepton decayed). Once these are
found and one or more are not a W boson, the mother particle (e.g. a b quark) is again
RECO Leptons
φ
Lepton pT GeV/c
90.660
0.294
(1) µ
63.470
0.314
(2) e
Lepton
(1) µ
(2) e
Quark
b
GEN Leptons
pT GeV/c
φ
89.290
37.590
0.294
0.314
GEN Quarks
pT GeV/c
φ
131.86
0.340
η
0.937
1.637
η
RECO
∆R : l(1)
RECO
∆R : l(2)
Mother
0.937
1.637
0.000
0.701
0.699
0.002
W
b
η
RECO
∆R : l(1)
RECO
∆R : l(2)
1.647
0.712
0.029
Table 5.2: Example of how the GEN-RECO matching for a high-tail MT 2 (ll) tt̄ + jets
event is done. First
p the RECO leptons are matched to the GEN leptons, based on
demanding ∆R = ∆φ2 + ∆η 2 ≈ 0. Then the PdgId of the GEN leptons is checked
to what the mother particle of the leptons is. If it is found that this is a b quark and
∆R(lRECO |b) < 0.06, the b quark is said to be matched to the lepton, indicating that
the lepton originated from a b-decay. The matched leptons and b quark are shown by
the interconnecting lines.
5
The PdgIds are the conventional numbering IDs for all elementary particles set by Particle Data
Group [51].
58
matched to its corresponding lepton based on the fact that their separation should not
be larger than ∆R = 0.06.
Of the 37 high-tail events, 29 out of 37 events have at least one selected lepton decayed
from a b quark and not a W boson and 8 out of 37 events are originating from W
decay. This means that ≈80% of MT 2 (ll) tail events are fakes; namely wrongly selected
leptons or fakes.
5.5
Reduction of the tt̄ + jets background
In order to reduce the fakes in the high tails of MT 2 (ll), an efficient way of making a
distinction between leptons originating from a W boson compared to a b quark decay
must be found. A first suggestion is to use the Reliso0X variable. It is constructed for
every selected lepton by drawing a circle of ∆R = 0.X in the angle space with center
around the lepton’s track and by computing
P
RelIso0X =
i∈all RECO particles
plep
T
piT
(5.3)
where i only runs over pT s that are inside the predefined circle as shown in Fig. 5.5 (a).
Since leptons that originate from b quarks will typically have lots of jet activity in their
vicinity, a RelIso0X < Y cut can reduce the contribution of those leptons. A rather
small cone (also referred to as MiniRelIso) does not help reducing the fakes in the
tails. The idea is that a lepton can be accidentally6 radiated away from the jet activity
(see Fig. 5.5 (b)) and a small cone cannot catch enough jet tracks for the event to fail
the cut. If one however adds a Reliso04 < 0.12 cut, where a larger cone size of 0.4 is
chosen, approximately 80% of fakes in the tails can be reduced [52]. The problem with
this approach, however, is that this also touches prompt leptons (those who originate
from W decay). It seems that the RelIso04 < 0.12 cut is not a good choice after all. A
new variable, that leaves the prompt leptons untouched, but removes specifically those
originating from semileptonic b quark decay, must be constructed.
6
This might seem to be unlikely, but we are looking at events in high tails of MT 2 (ll), so we expect
things to be unlikely!
59
(a)
(b)
(c)
Figure 5.5: Display of the RelIso0X approach with conesize 0.X for (a) a lepton near
jet activity (usual case), (b) a lepton radiated away from jet activity with conesize 0.X
and (c) a lepton ejected away from jet activity with a conesize 0.Y > 0.X.
This new approach is to construct the MultiIsoV2 variable, defined as
MultiIsoV2 = MiniRelIso < A && (ptRatioV2 > B k ptRelV2 > C) .
(5.4)
It contains three components, explained below:
• MiniRelIso: This is the same principle as RelIso, but corrected for boosts. The
cone size gets a pT dependence: the higher the pT of the lepton is, the smaller
the cone size necessary to catch jet activity as the jet will have a much smaller
opening angle.
Jet
• ptRatioV2: It is defined as plep
T /pT , where the closest jet to the lepton is considered. If the lepton has a significantly larger pT than the jet, chances are small it
originated from the b quark.
• ptRelV2: It is defined as k~pTlep − p~Jet
T k and is displayed in Fig. 5.6a. The larger
this value, the further the lepton will be separated from the jet, so the smaller
the chances are it originated from the b quark.
60
(a) Display of how ptRelV2 is computed.
(b) Approximately 80% of fake high tail tt̄ +
jets events are killed by the MultiIsoV2 cut.
Figure 5.6
The MultiIsoV2 cut acting on the tt̄ + jets background is displayed in Fig. 5.7. Here,
A, B, C were taken to be respectively 0.09, 0.84, 7.2. It’s clear how the low MT 2 (ll)
region gets reduced by about 15%, but from about 110 GeV/c2 onwards, a clear reduction of tail events becomes visible.
Events / 10 GeV/c2
MultiIsoV2 = miniRelIso(l ,l ) < 0.09 && (jetPtRatiov2(l ,l ) > 0.84 || jetPtRelv2(l ,l ) > 7.2)
1 2
1 2
MultiIsoV2 thus effectively reduces fakes in
the high MT 2 (ll) region (≈80%, see Fig.
5.6b).
Over the entire MT 2 (ll) spectrum,
the tt̄ + jets background
is reduced by approximately 48% and the signal only loses about 4%
[52].
1 2
tt + Jets (lep)
103
tt + Jets (lep) + MultiIsoV2
2
10
10
1
10− 1
10− 2
1.20
1
0.8
0.6
0.4
0.2
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
MT2(ll)
Figure 5.7: Effect of the MultiIsoV2 cut on the tt̄ + jets
background in the MT 2 (ll) distribution before the cut (cyan)
and after the cut (red).
61
5.6
Summary and Discussion
In this section, an analysis is presented within the scope of an opposite sign SUSYsearch for stop quarks, with the production process depicted in Fig. 5.1. An overview
of the analysis strategy is given along with the most important cuts that need to be applied in order to reduce background processes as much as possible. Their effect can be
seen in the cutflow-diagram presented in Table 5.1. The various backgrounds that have
to be taken into account are presented with their corresponding Feynman diagrams on
page 56.
It is noted that the tt̄ + jets background requires extra attention. This can be seen
from Fig. 5.4 and also from the cutflow-diagram in Table 5.1 where tt̄ + jets events
appear in the high tails of MT 2 (ll) (> 140 GeV/c2 ). As is explained in Sec. 5.2, events
originating from a SM tt̄ process shouldn’t be found for high values of MT 2 (ll). To find
which selected leptons are present in the high-tails, GEN information is compared to
RECO information to identify the mother particles of the selected leptons. An example
of this is given in Table 5.2. It is found that 29 out of 37 events in the tails have leptons
originating from a b quark instead of a W boson. This means that for those events, the
wrong leptons are selected, and should be categorized as fakes.
Two approaches are suggested to reduce the fakes in the tails. The first, adding a
RelIso04 < 0.12 cut, does not turn out to be efficient as also prompt leptons are
touched by it. This is presented in [52]. Therefore, a MultiIsoV2 cut is added instead
as defined in Eq. 5.4. This cut seems to be very efficient in reducing the tails and leaves
the signal untouched. The tt̄ + jets background is reduced by 48% in total with a loss
of 80% in the tails and the signal is only reduced by 4% in the tails. This can also be
seen from the pie chart in Fig. 5.6b (b) and Fig. 5.7.
62
Chapter 6
Analysis Part II: Parton Shower
Tuning in Modelling Top Quark
Pairs
6.1
Introduction
MC event generators in high energy physics employ various physics models to describe
a large variety of different types of processes [53]. Their results are used to devise search
strategies or background estimates of physical processes to allow theoretical interpretations of experimental results. MC generators should provide a good description of
available data, providing confidence in the theoretical models underlying the generator
performance. Due to the large amount of data available from the LHC and previous experiments, experimental uncertainties have become more precise. This allows detailed
studies on the validity of MC simulations and tuning (Sec. 4.3.3) efforts pave the
way towards new and more precise predictions. The free parameters of these models
need to be tuned from distributions of suitable observables extracted from experimental
measurements.
It has been known that the Njets observable as depicted in Fig. 6.1 is not well simulated
by PYTHIA8, causing inaccurate MC outputs. Overestimation of MC events (generated
with POWHEG+PYTHIA8) with respect to data becomes visible for larger Njets bins. This
has also been observed with 13 TeV data (in the dilepton channel [54] and the single
lepton channel [55]) as well as with MADGRAPH5 aMC@NLO+PYTHIA8 in both FxFx and
MLM configurations1 . Among others, two main causes for this are identified: (1) The
1
Those are two merging schemes used in MADGRAPH5 aMC@NLO for NLO and LO cases respectively,
63
matching scheme between POWHEG and PYTHIA8, or (2) some of the PYTHIA8 parameters
are not optimal, where a combination of both issues might be a possibility too. In this
particular study, an effort is made to explore whether solutions can be found using
dedicated tools to perform a tune on one of the parameters in PYTHIA8.
To tackle this issue, αSISR , a PYTHIA8 parameter that controls Initial State Radiation
(ISR) effects in the parton shower process could be tuned [56]. The tuning tool used
is Professor [47], a program for tuning model parameters of MC event generators to
experimental data by parametrising the per-bin generator response to parameter variations and numerically optimising parametrised behaviour. More information on the
functioning of Professor can be found in Appendix A.
The POWHEG event generator provides a matrix element calculation of the pp → tt̄ up
to an additional jet (not present in the LO calculation) matched to parton shower MC
programs (in our case PYTHIA8).
19.7 fb−1 (8 TeV)
1/σvis dσvis /dNjets
CMS
b
b
10−1
b
Data
POWHEG+PYTHIA8
b
b
Theory/Data
10−2
b
1.4
1.2
1
0.8
0.6
2
3
4
5
6
Njets [ p T > 30 GeV]
Figure 6.1: Normalised
√ tt̄ Cross-section in the dilepton channel in function of Njets
(pT > 30 GeV) for s = 8 TeV (black) and the POWHEG+PYTHIA8 MC generator
output (red).
see Sec. 4.3.2 for a brief discussion on merging.
64
6.2
Tuning of αSISR
In this study, we tune the SpaceShower:alphaSvalue parameter2 of PYTHIA8, here denoted by αSISR . 13 MC runs are generated for different values of αSISR . The used values
are listed in Table 6.1.
Run
0
1
2
3
4
5
6
αSISR
0.100
0.105
0.110
0.115
0.120
0.125
0.130
Run
7
8
9
10
11
12
αSISR
0.135
0.140
0.145
0.020
0.050
0.080
Table 6.1: Tabular overview of 13 MC runs for different values of αSISR .
For runs 0-9, αSISR ∈ [0.100, 0.145] is chosen, since it is expected that αSISR would lie
in between these bounds. For runs 10-12 however, smaller values are selected to make
sure the tuning would not navigate towards them, which would clearly be a wrong result.
Briefly, the Professor method goes as follows. A MC response function M Cb (p) is
constructed for every observable’s bin b. Afterwards, for every point p in parameter
space, a χ2 function is constructed according to
χ2 (p) =
X X (M Cb (p) − Rb )
O b∈O
∆2b
,
(6.1)
where O denotes the observable, Rb the data at bin b, and ∆2b the total uncertainty
originating from the quadrature sum of MC-uncertainty and data-uncertainty. Note
that this formula corresponds to Eq. A.7 from Chapter A, with the weights wi chosen
equal to 1. For every parameter point p, the χ2 is then interpolated and eventually
ISR
minimised. The value of p corresponding to a minimal χ2 , pbest (here αS,best
) is the
final tune.
This study selects four observables to which a tune can be performed. They are Njets
for pT > 30, 60 and 100 GeV/c and subleading additional3 jet pT . Varying the values of
αSISR for different MC-runs yields a different output in each observable bin. The outliers
2
For a full list of the PYTHIA8 parameters, refer to [57].
The subleading additional jet denotes the fourth most energetic jet in the dilepton channel. The
two b-jets and one ISR-jet are produced in POWHEG (at NLO) and are not sensitive to αSISR .
3
65
of the αSISR variations are called envelopes and are displayed in Figs. 6.2 (a), (b), (c)
and (d).
Of the four observables, three are used to conduct three tuning efforts. They are listed
below.
1
Njets [pT > 30 GeV]
2
Njets [pT > 60 GeV]
3
Njets [pT > 30 GeV] and Subleading additional jet pT
Table 6.2: Different observables used for tuning. They are also displayed in Fig. 6.2.
For each of the three tuning efforts, Professor performs a tune towards the observables
mentioned in Table 6.2. In case of the Njets -observables, only bins with Njets > 3 are
considered in the tuning process, where jets predominantly originate from the parton
shower and hence these bins are sensitive to αSISR . Note that the distribution Njets (pT >
100 GeV) displayed in Fig. 6.2 (c) is not used in either of the combinations given in
Table 6.2. The CMS analysis code used is from TOP-12-041 [58] along with the Rivet
routine CMS 2015 I1397174.
66
19.7 fb−1 (8 TeV)
Envelope
Data
b
b
b
10−1
19.7 fb−1 (8 TeV)
CMS
1/σvis dσvis /dNjets
1/σvis dσvis /dNjets
CMS
b
b
b
Envelope
Data
b
10−1
b
b
b
b
10−2
b
b
1.4
Theory/Data
Theory/Data
10−2
1.2
1
0.8
0.6
2
3
4
1.4
1.2
1
0.8
0.6
5
6
Njets [ p T > 30 GeV]
0
19.7 fb−1 (8 TeV)
CMS
1
2
1/σvis dσvis /dNjets
b
b
b
Envelope
Data
b
10−1
b
10−2
19.7 fb−1 (8 TeV)
b
10−2
b
b
Envelope
Data
b
10−3
b
10−4
b
b
1.4
Theory/Data
Theory/Data
4
5
Njets [ p T > 60 GeV]
(b)
CMS
1/σvis dσvis /dp T [GeV−1 ]
(a)
3
1.2
1
0.8
0.6
0
1
2
3
4
Njets [ p T > 100 GeV]
10−5
1.4
1.2
1
0.8
0.6
50
(c)
100
150
200
250
300
350
400
Subleading additional jet p T [GeV]
(d)
Figure 6.2: Envelope-plots for the four different possible observables that can be used
for tuning. The blue bands denote how MC variations in αSISR from Table 6.1 overlap
with the data. The yellow bands display the data uncertainties.
67
6.3
Estimation of uncertainty on αISR
S
Uncertainties on the tuned value of αSISR are estimated in two ways. The first method
(1) is to consider the data uncertainties. For each observable combination, two additional datasets are constructed by varying the central data up with the upper error,
and varying it down with the lower error. After this, the three datasets are used to
perform three tunes. The resulting tune using the central dataset is the central tune
result. The ones resulting from the upwards scaled and downwards scaled datasets are
the respective upper and lower bounds of the central tune results4 .
The second method (2) is to construct two eigentunes for each tune result5 . The
parameter-space used here is one-dimensional and thus the eigentunes are found by
varying the tuned result up and down with the value of the χ2 (this is the standard
Professor -prescription). The values of αSISR corresponding to those variations are then
its upper and lower bounds. The results can eventually be displayed for the studied observables alongside the central tune. A more thorough explanation on the construction
of eigentunes is given in Appendix A.3.
6.4
Results
Result
αSISR (central)
1
0.115
2
0.104
3
0.113
Uncertainties (1)
+0.021
-0.019
+0.014
-0.012
+0.029
-0.021
Percentages
18.635%
19.048%
13.114%
11.555%
25.402%
18.296%
Table 6.3: of the tuning results taking the three different tunings as listed in Table 6.2
as input.
The result of each tuning is summarised in Table 6.3. For each tuning one, the central
tuning result is given along with its uncertainties according to method (1) described
in Sec. 6.3, both in absolute values as percentages. The Professor output from taking
the eigentunes into account, according to method (2), are omitted for this set of tuning
4
Note that for the upwards and downwards scaled datasets, the uncertainties in data are kept
constant.
5
This can be done using the –eigentunes option in the prof-tune command.
68
efforts. The reason for this is that the tuning is done to only one or two distributions
at most for typically a low amount of degrees of freedom. Therefore the resulting χ2
is small and varying up the χ2 by itself to acquire the errors yields unrealistically low
uncertainties on the tuned value of αSISR .
The results are displayed in Figs. 6.4, 6.5 and 6.6. The resulting tunes are displayed in Fig 6.3, along with the ATLAS ATTBAR-POWHEG tune [59], the PYTHIA8
(CUETP8M1) and PYTHIA6 (CTEQ5M) defaults. Due to the large data-uncertainties
on the subleading additional jet pT observable, it isn’t taken into account for the further
remaining part of the study. The optimal tuning result is chosen to be combination
1 in which the tuning towards the Njets (pT > 30 GeV) distribution was performed
as its result lies well within experimental data-uncertainties independent of the jet pT
treshold. The value of αSISR = 0.115+0.021
−0.019 is thus considered to be the optimal tune.
CMS
Tune 1
30
021
0.115−+00..019
CMS
Tune 2
60
014
0.104−+00..012
CMS
Tune 3
30
029
0.113−+00..021
Njet
Njet
Njet + sub leading add. jet pT
ATLAS ATTBAR-POWHEG
ATL-PHYS-PUB-2015-007 (7 TeV data)
0.121
CUETP8M1
0.137
CTEQ5M
0.127
0.050
0.100
0.150
αSISR
Figure 6.3: Summary of tune results and their uncertainties. Two alternative tunes
are presented in addition to the previously described one. The input distribution(s)
for each tune is indicated below the label of each tune. The αSISR values in the ATLAS
ATTBAR-POWHEG tune for PYTHIA8, the CMS CUETP8M1 tune for PYTHIA8 (CMS
default in Run II), and the CMS Z2* PYTHIA6 tune (CMS default in Run I), using the
CTEQ5M PDF set, are also displayed.
69
1/σvis dσvis /dNjets
1/σvis dσvis /dNjets
b
b
10−1
b
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
b
b
b
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
dummy
b
POWHEG+PYTHIA8 CUETP8M1
Theory/Data
1.4
1.2
1
0.8
0.6
2
3
4
b
POWHEG+PYTHIA8 CUETP8M1
10−2
αISR
S = 0.115
αISR
S = 0.096
ISR
αS = 0.137
10−3
Theory/Data
10−1
dummy
10−2
b
b
αISR
S = 0.115
αISR
S = 0.096
ISR
αS = 0.137
b
1.4
1.2
1
0.8
0.6
5
6
Njets [ p T > 30 GeV]
0
1
2
b
b
b
10−1
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
POWHEG+PYTHIA8 CUETP8M1
αISR
S = 0.115
αISR
S = 0.096
αISR
S = 0.137
b
dummy
10−1
POWHEG+PYTHIA8 CUETP8M1
b
10−2
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
αISR
S = 0.115
αISR
S = 0.096
αISR
S = 0.137
b
b
10−3
b
b
10−3
1.4
1.2
1
0.8
0.6
0
1
10−4
b
Theory/Data
Theory/Data
b
dummy
10−2
4
5
Njets [ p T > 60 GeV]
(b)
1/σvis dσvis /dp T [GeV−1 ]
1/σvis dσvis /dNjets
(a)
3
1
2
3
4
Njets [ p T > 100 GeV]
10−5
1.4
1.2
1
0.8
0.6
50
(c)
100
150
200
250
300
350
400
Subleading additional jet p T [GeV]
(d)
Figure 6.4: Results of αSISR tuning 1. Jet multiplicity distribution, Njets (with pjet
T > 30
GeV), after tuning αSISR with the Njets > 3 bins (where jets predominantly originate from
the parton shower) is used as input to Professor [47]. The unfolded CMS data are shown
with total error bars. In each plot, the calculated distribution assuming the tuned αSISR
is shown (solid line). The calculated distributions with the lower bound (dashed line)
and the upper bound (dot-dashed line) of the tuned αSISR are also displayed. Beneath
each plot is shown the ratio of theory predictions to data. The yellow bands indicate
the total data uncertainty.
70
1/σvis dσvis /dNjets
1/σvis dσvis /dNjets
b
b
10−1
b
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
b
b
b
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
dummy
b
POWHEG+PYTHIA8 CUETP8M1
Theory/Data
1.4
1.2
1
0.8
0.6
2
3
4
b
POWHEG+PYTHIA8 CUETP8M1
10−2
αISR
S = 0.104
αISR
S = 0.092
ISR
αS = 0.117
10−3
Theory/Data
10−1
dummy
10−2
b
b
αISR
S = 0.104
αISR
S = 0.092
ISR
αS = 0.117
b
1.4
1.2
1
0.8
0.6
5
6
Njets [ p T > 30 GeV]
0
1
2
b
b
b
10−1
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
POWHEG+PYTHIA8 CUETP8M1
αISR
S = 0.104
αISR
S = 0.092
αISR
S = 0.117
b
dummy
10−1
POWHEG+PYTHIA8 CUETP8M1
b
10−2
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
αISR
S = 0.104
αISR
S = 0.092
αISR
S = 0.117
b
b
10−3
b
b
10−3
1.4
1.2
1
0.8
0.6
0
1
10−4
b
Theory/Data
Theory/Data
b
dummy
10−2
4
5
Njets [ p T > 60 GeV]
(b)
1/σvis dσvis /dp T [GeV−1 ]
1/σvis dσvis /dNjets
(a)
3
1
2
3
4
Njets [ p T > 100 GeV]
10−5
1.4
1.2
1
0.8
0.6
50
(c)
100
150
200
250
300
350
400
Subleading additional jet p T [GeV]
(d)
Figure 6.5: Results of αSISR tuning 2. Jet multiplicity distribution, Njets (with pjet
T > 60
GeV), after tuning αSISR with the Njets > 3 bins (where jets predominantly originate from
the parton shower) is used as input to Professor [47]. The unfolded CMS data are shown
with total error bars. In each plot, the calculated distribution assuming the tuned αSISR
is shown (solid line). The calculated distributions with the lower bound (dashed line)
and the upper bound (dot-dashed line) of the tuned αSISR are also displayed. Beneath
each plot is shown the ratio of theory predictions to data. The yellow bands indicate
the total data uncertainty.
71
1/σvis dσvis /dNjets
1/σvis dσvis /dNjets
b
b
10−1
b
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
b
b
b
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
dummy
b
POWHEG+PYTHIA8 CUETP8M1
Theory/Data
1.4
1.2
1
0.8
0.6
2
3
4
b
POWHEG+PYTHIA8 CUETP8M1
10−2
αISR
S = 0.133
αISR
S = 0.093
ISR
αS = 0.142
10−3
Theory/Data
10−1
dummy
10−2
b
b
αISR
S = 0.133
αISR
S = 0.093
ISR
αS = 0.142
b
1.4
1.2
1
0.8
0.6
5
6
Njets [ p T > 30 GeV]
0
1
2
b
b
b
10−1
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
b
POWHEG+PYTHIA8 CUETP8M1
αISR
S = 0.133
αISR
S = 0.093
αISR
S = 0.142
b
dummy
10−1
POWHEG+PYTHIA8 CUETP8M1
b
10−2
CMS data 19.7fb−1 (8 TeV)
arXiv:1510.03072
Data Uncertainty
αISR
S = 0.133
αISR
S = 0.093
αISR
S = 0.142
b
b
10−3
b
b
10−3
1.4
1.2
1
0.8
0.6
0
1
10−4
b
Theory/Data
Theory/Data
b
dummy
10−2
4
5
Njets [ p T > 60 GeV]
(b)
1/σvis dσvis /dp T [GeV−1 ]
1/σvis dσvis /dNjets
(a)
3
1
2
3
4
Njets [ p T > 100 GeV]
10−5
1.4
1.2
1
0.8
0.6
50
(c)
100
150
200
250
300
350
400
Subleading additional jet p T [GeV]
(d)
Figure 6.6: Results of αSISR tuning 3. Jet multiplicity distribution, Njets (with pjet
T > 30
GeV) and Subleading additional jet pT , after tuning αSISR with the Njets > 3 bins (where
jets predominantly originate from the parton shower) are used as input to Professor [47].
The unfolded CMS data are shown with total error bars. In each plot, the calculated
distribution assuming the tuned αSISR is shown (solid line). The calculated distributions
with the lower bound (dashed line) and the upper bound (dot-dashed line) of the tuned
αSISR are also displayed. Beneath each plot is shown the ratio of theory predictions to
data. The yellow bands indicate the total data uncertainty.
72
6.5
The renormalisation scale µR
From Sec. 6.4, upper and lower bound uncertainties for the optimal αSISR were found.
It is also possible however, to vary αSISR by altering the renormalisation scale µR . Up
to one loop correction, αS is given by
αS ≈
1
2 µ
β(n) ln Λ2 R
(6.2)
QCD
33 − 2n
(6.3)
with β(n) =
12π
with n being the number of active quark flavours and ΛQCD the QCD-scale. Using this
relation between αS and µR , we can determine the scale variations corresponding to
the uncertainties found for αSISR . This is displayed in Fig. 6.7. In Eq. 6.2, ΛQCD is
first determined for αS set to the central value and µR set to 91.2 GeV (Z boson mass).
Then the upper and lower bounds for αS are filled in and it is checked how much µR
has to be varied for a fixed ΛQCD . The variations are displayed in the legend of Fig.
6.7 and correspond to 0.33 for the upper bound and 4.10 for the lower bound.
αS(mZ) = 0.115
αS(mZ) = 0.096
αS(mZ) = 0.137
0.160
αS
0.140
0.120
0.100
0.080
101
0.33 · mZ
mZ
102
µR (GeV)
4.10 · mZ
103
Figure 6.7: αS as a function of the renormalization scale, µR . The uncertainty on the
tuned αS value (0.115) corresponds to variations of µR by factors of 0.33 for the upper
bound and 4.10 for the lower bound.
73
6.6
Comparison between Professor Interpolation and
MC Output
19.7 fb−1 (8 TeV)
1/σvis dσvis /dNjets
CMS
b
b
10−1
b
Data
αISR
S = 0.115 (Interpolation)
αISR
S = 0.115 (MC)
b
Theory/Data
10−2
b
b
1.4
1.2
1
0.8
0.6
2
3
4
5
6
Njets [ p T > 30 GeV]
Figure 6.8: Comparison between the Professor interpolation and the MC output for
tuning result 1 with αSISR = 0.115 for the Njets (pT > 30 GeV) distribution. The interpolated results from Professor (red) nicely overlap with the MC data (blue) generated
for the same value of αSISR . The yellow bands denote the data uncertainties.
The Professor tuning result 1 is given by αSISR = 0.115 and is used as input to a
POWHEG+PYTHIA8 MC run. The Professor interpolation and MC output are compared
for the Njets (pT > 30 GeV) observable as shown in Fig. 6.8. It is to be seen that
both results are nicely in agreement with each other, which provides confidence that
Professor indeed produces correctly interpolated observable distributions.
74
6.7
Validation with other Top Quark Distributions
The studied tunes are validated with the TOP-15-011 distributions (with Rivet routine
CMS 2015 I1370682) [60]. Concretely, it is checked using tune results 1, 2 and 3 (see
Fig. 6.3) what the MC output with respect to data is for the observables ptT , ∆φtt̄ , ptTt̄
and mtt̄ in the single lepton and dilepton channel.
In the single lepton and dilepton channel, we can see from Figs. 6.9 (a), (b), (c) and
(d) that there is a good agreement between the different αSISR values. However, as expected, the ptT remains unchanged and is still not well described by the MC6 . Figs 6.10
(a) and (b) show how for the ptTt̄ and mtt̄ distributions higher values of αSISR are slightly
preferred over tune 1. From Figs. 6.10 (c) and (d), the variations in αSISR in the ptTt̄ and
mtt̄ distributions do not seem to have a significant effect on the MC output.
6
The incorrect description of top pT is known to be caused by missing higher order effects.
75
tt̄ → bblνjj | 19.7 fb−1 (8 TeV)
0.007
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
b
0.006
0.005
b
0.004
b
0.003
tt̄ → bblνjj | 19.7 fb−1 (8 TeV)
CMS
(1/σ ) dσ/d∆φ
(1/σ ) dσ/dpT [GeV−1 c]
CMS
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
1
b
b
b
b
0.002
10−1
0.001
b
b
b
b
Theory/Data
Theory/Data
0
1.4
b
1.2
1
0.8
0.6
0
100
200
300
1.4
1.2
1
0.8
0.6
400
500
pTt [GeV/c]
0
0.5
1
1.5
(a)
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
b
0.006
0.005
b
0.004
0.003
b
0.002
b
1
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
b
b
10−1
0.001
b
b
b
0
1.4
Theory/Data
Theory/Data
3
∆φtt̄ [rad]
tt̄ → bblνlν | 19.7 fb−1 (8 TeV)
CMS
(1/σ ) dσ/d∆φ
(1/σ ) dσ/dpT [GeV−1 c]
0.007
2.5
(b)
tt̄ → bblνlν | 19.7 fb−1 (8 TeV)
CMS
2
1.2
1
0.8
0.6
0
50
100
150
200
250
300
350
400
pTt [GeV/c]
(c)
1.4
1.2
1
0.8
0.6
0
0.5
1
1.5
2
2.5
3
∆φtt̄ [rad]
(d)
Figure 6.9: Validation of the TOP-15-011 distributions for the variables ptT and ∆φtt̄ in
the single lepton ((a) and (b)) and the dilepton ((c) and (d)) channels for the four different tune results. The ATLAS ATTBAR-POWHEG tune and the PYTHIA8 (CTEQ5M)
and PYTHIA6 (CUETP8M1) defaults are also displayed.
76
tt̄ → bblνjj | 19.7 fb−1 (8 TeV)
(1/σ ) dσ/dpTtt̄ [GeV−1 c]
0.016
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
0.014
b
0.012
b
0.01
0.008
b
0.006
tt̄ → bblνjj | 19.7 fb−1 (8 TeV)
CMS
(1/σ ) dσ/dmtt̄ [GeV−1 c2 ]
CMS
0.004
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
10−2
b
b
b
10−3
b
b
10−4
b
b
0.002
10−5
b
b
Theory/Data
Theory/Data
0
1.4
1.2
1
0.8
0.6
0
50
100
150
200
b
1.4
1.2
1
0.8
0.6
250
300
pTtt̄ [GeV/c]
400
600
800
1000
(a)
b
b
0.014
0.012
0.01
0.008
b
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
= 0.113 (CMS Tune 3)
αISR
S
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
0.006
tt̄ → bblνlν | 19.7 fb−1 (8 TeV)
CMS
(1/σ ) dσ/dmtt̄ [GeV−1 c2 ]
(1/σ ) dσ/dpTtt̄ [GeV−1 c]
0.016
1400
1600
mtt̄ [GeV/c2 ]
(b)
tt̄ → bblνlν | 19.7 fb−1 (8 TeV)
CMS
1200
b
10−2
b
b
b
10−3
b
10−4
Data
= 0.115 (CMS Tune 1)
αISR
S
= 0.104 (CMS Tune 2)
αISR
S
ISR
αS = 0.113 (CMS Tune 3)
= 0.121 (ATLAS Tune)
αISR
S
= 0.127 (CUETP8M1)
αISR
S
= 0.137 (CTEQ5M)
αISR
S
b
0.004
10−5
b
0
1.4
b
b
Theory/Data
Theory/Data
0.002
1.2
1
0.8
0.6
0
50
100
150
200
250
300
pTtt̄ [GeV/c]
(c)
1.4
1.2
1
0.8
0.6
400
600
800
1000
1200
1400
1600
mtt̄ [GeV/c2 ]
(d)
Figure 6.10: Validation of the TOP-15-011 distributions for the variables ptTt̄ and mtt̄ in
the single lepton ((a) and (b)) and the dilepton ((c) and (d)) channels for the four different tune results. The ATLAS ATTBAR-POWHEG tune and the PYTHIA8 (CTEQ5M)
and PYTHIA6 (CUETP8M1) defaults are also displayed.
77
6.8
Summary and Discussion
In this study, the PYTHIA8 parameter αSISR is tuned using Njets -distributions. The optimal value is chosen to be αSISR = 0.115+0.021
−0.019 as its correspondence with data is the
best as can be seen from Figs. 6.4 (a), (b), (c) and (d) and it has the lowest uncertainties. Furthermore, within uncertainties it is in agreement with the ATLAS ATTBARPOWHEG tune (Fig. 6.3 and ref. [59]). The uncertainties seem to correspond to a
factor 0.33 and 4.10 for the respective upper and lower bounds of the renormalisation
scale µR . The different tunes give αSISR of the order 0.10-0.11, however when applying
these to TOP-15-011 distributions (Figs 6.9 and 6.10), the value of αSISR = 0.1273 seems
to give a slightly better description. For further tests, the tt̄ samples for ICHEP 2016
have been submitted with αSISR = 0.1273, with the same value as the PYTHIA6 default
(CTEQ5) used in Run I CMS simulations.
CUETP8M1 predictions using underlying event Drell-Yan pT and rapidity y, are tested
with the tuned αSISR values. No significant changes have been observed [61]. The main
differences have been observed in underlying event observables up to ∼10-15% [61].
This shows that it is desirable to have a new underlying event tune that has a lower
αSISR value.
Tuning αSISR in POWHEG+PYTHIA8 is the most straightforward approach to fix the discrepancies seen in the Njets distributions (e.g. Fig. 6.1). This process however may
not be as simple in MADGRAPH5 aMC@NLO+PYTHIA8 since for this case the αSISR must be
modified in the matrix element as well. The tune found makes the agreement of 13 TeV
data with POWHEG+PYTHIA8 as good as 8 TeV with MADGRAPH5+PYTHIA6, i.e. a very
good description of all distributions (in the considered low pT and mass ranges) except
for ptT , which is due to missing higher order effects. Note also that parton shower αSISR
tuning should depend on the QCD scale choice and on the hdamp parameter in POWHEG
(which is taken to be hdamp = mt ). In a finer tuning attempt hdamp and αSISR should
be tuned simultaneously or consecutively. For instance, one can first tune αSISR as has
been done in this study and then tune the hdamp variable using the first additional jet
pT distribution as input.
To conclude with, the description of Njets is fixed using a lower αSISR value than the
PYTHIA8 default of 0.1365. This change does not cause significant alterations and
therefore with the lower αSISR value all top quark distributions except ptT are described
well by POWHEG+PYTHIA8. These results are also made public as additional material to
[58].
78
Hoofdstuk 7
Nederlandstalige Samenvatting
7.1
Introductie
Het Standaardmodel (SM) van de elementaire deeltjesfysica is een zeer succesvolle theorie die in staat is om een erg rijk gamma aan experimenten te beschrijven. Het beschrijft
de drie fundamentele krachten van het elektromagnetisme, de zwakke kernkracht en de
sterke kernkracht, zoals beschreven in Sectie 1.2. De vierde fundamentele kracht, nl. de
zwaartekracht, past niet binnen dit framework, vermits het SM niet compatibel is met
de algemene relativiteitstheorie. Daarnaast levert het een classificatie van alle fundamentele deeltjes die tot op heden gekend en waargenomen zijn. Deze worden beschreven
in Sectie 1.1.
Het waarnemen van deze fundamentele deeltjes gebeurt d.m.v. een groot aantal experimenten verspreid over de ganse wereld. In deze studie wordt toegespitst op het
CMS (Compact Muon Solenoid ) experiment (hoofdstuk 3), uitgevoerd in CERN in
Genève, Zwitserland. CMS is een detector die geı̈nstalleerd is op de LHC (Large Hadron Collider ), waarmee protonen versneld worden tot enorme snelheden en botsen op
welbepaalde interactiepunten aan hoge massacentrumenergieën. CMS is cilindrisch van
vorm en bevindt zich rond een dergelijk interactiepunt waar het tracht de vervalproducten van de botsingen te detecteren.
Hoewel het SM erg succesvol is, bestaan er toch vele tekortkomingen en fenomenen die
niet verklaard kunnen worden zonder een uitbreiding van de theorie. Enkele van deze
tekortkomingen worden gegeven in Sectie 1.4. Een erg populaire theorie die het SM
uitbreidt is Supersymmetrie (SUSY), zoals uiteengezet in hoofdstuk 2. SUSY postuleert het bestaan van een superpartner voor alle reeds gekende fundamentele deeltjes.
79
Hier wordt het Minimale Supersymmetrische Model (MSSM) aangenomen. Dit betreft
een minimale uitbreiding van het SM en veronderstelt het behoud van R-pariteit. Deze
symmetrie zorgt ervoor dat supersymmetrische deeltjes steeds in paren geproduceerd
worden en dat er een lichtst supersymmetrisch deeltje (LSP) bestaat, het neutralino χ01 ,
dat bovendien stabiel is. Bovenstaande inzichten worden in Sectie 2.3 in meer detail
uiteengezet. Een SUSY-onderzoek dat in deze thesis gedaan wordt, staat hieronder in
Sectie 7.2 beschreven.
Naast de uitbreiding van het SM d.m.v. SUSY, is het in deze thesis ook belangrijk om
stil te staan bij Monte Caro (MC) generatoren. Deze zijn onmisbaar bij hoge energie
fysica experimenten. MC generatoren bestaan uit een set van modellen die instaan voor
het simuleren van evenementen zoals deze die optreden bij botsingen tussen hadronen
(of hier, botsingen tussen protonen). Wij spitsen ons toe op twee aparte types van MC
generatoren. Ter illustratie bekijken wij POWHEG en PYTHIA.
POWHEG is een Matrix Element (ME) MC generator. Dit betekent dat deze instaat voor
het genereren van het initiële hard scatter proces waarbij een hoge momentumoverdracht plaatsvindt. Dit is m.a.w. de interactie tussen twee partonen uit elk van beide
protonen. Powheg berekent dit proces tot op Next-to-Leading-Order (NLO), waarbij
het dus een harde partonenemissie in rekening neemt, hetgeen weergegeven wordt in
het Feynman diagram in Fig. 4.2.
PYTHIA is een MC generator die hoge energie fysica evenementen genereert. Dit zijn sets
van deeltjes die geproduceerd werden uit de interactie tussen twee inkomende deeltjes.
Het bevat theoretische modellen die de evolutie van een proces met hoge momentumoverdracht simuleren tot een complexe multi-hadronische finale toestand. Deze evolutie
wordt Parton Shower (PS) genoemd. Een parton verliest immers steeds momentum
tijdens diens propagatie doorheen de ruimte (d.m.v. de interacties met quarks en gluonen). Eens een bepaalde drempelwaarde overschreden wordt, treedt hadronisatie op
waarbij kleurloze finale toestanden gevormd worden.
Aangezien MC generatoren tal van benaderingen gebruiken, worden parameters scherpgesteld om er voor te zorgen dat de predicties overeenkomen met wat gemeten wordt
in data. Dit proces heet tuning en wordt gedaan met specifieke programma’s. Hiertoe
kan Professor gebruikt worden, een tuning code die in meer detail beschreven staat in
Appendix A. Een concrete toepassing van het scherpstellen van een PYTHIA8 parameter,
nl. αSISR , wordt uiteengezet in Sectie 7.3, verder in deze tekst.
80
7.2
Deel I: Zoeken naar scalaire top quark partners
Deel I van deze studie is een deelonderzoek in een SUSY-analyse m.b.t. het
√zoeken naar
de scalaire superpartner van de top quark bij een massacentrumenergie s = 13 TeV
voor proton-proton botsingen. Het gepostuleerde proces wordt gegeven in Figuur 5.1,
waarbij specifiek wordt gezocht naar het kanaal waarin beide W bosonen leptonisch
vervallen.
Het is de bedoeling bij dergelijke studies dat een signaalregio geconstrueerd wordt.
Dit is een bepaald stuk van de faseruimte waar het signaal de achtergrondprocessen
gegeven op blz. 56 domineert. De discriminerende parameter die hiervoor gebruikt
wordt is MT 2 (ll) (gedefinieerd door Vgl. 5.1). Het is uiteraard belangrijk dat in de
signaalregio de achtergronden, die berekend zijn door Monte Carlo generatoren, correct
opgenomen zijn. Zoals aangegeven in Tabel 5.1 en Figuur 5.4, wordt opgemerkt dat de
tt̄ + jets achtergrond toch voor hoge waarden van MT 2 (ll) (> 140 GeV/c2 ) aanwezig is.
Voor een dergelijk SM proces is dit onmogelijk. Een verklaring en oplossing voor het
optreden van tt̄ + jets evenementen in dit gebied wordt vervolgens uit de doeken gedaan.
In de regio MT 2 (ll) > 140 GeV/c2 worden de aanwezige tt̄ + jets evenementen specifiek
uitgezocht. De gereconstrueerde kinematische informatie (RECO) wordt gelinkt aan de
gegenereerde kinematische informatie (GEN) om te achterhalen wat de moederdeeltjes
zijn van de geselecteerde leptonen. Een voorbeeld hiervan wordt gegeven in Sectie 5.4.
Van de 37 gevonden evenementen blijken er 29 afkomstig van b quark verval en niet
van leptonisch W boson verval. Dit betekent dat een deel van de tt̄ + jets evenementen
verkeerdelijk geselecteerd worden in de analyse.
Voor dit probleem worden twee oplossingen vooropgesteld. Ten eerste wordt gekeken
of een RelIso04 < 0.12 cut het probleem zou oplossen. De motivatie voor deze keuze
is om de grootte van de fictieve kegel getekend rondom het geselecteerde lepton te vergroten om zo meer jet activiteit op te vangen en zo leptonen van b quark verval (dicht
bij jet activiteit) te verwijderen. Dit kan eenvoudig begrepen worden uit Vgl. 5.3 en
Figuur 5.5. Het effect hiervan was dat deze methode wel degelijk ∼80% van de verkeerdelijk geselecteerde tt̄ + jets evenementen reduceert, maar dit echter ook inefficiënt
is voor het signaal. De methode bestaat er in om een MultiIsoV2 cut toe te passen,
zoals gedefinieerd in Vgl. 5.4. Deze heeft het gewenste effect om wederom ∼80% van
de verkeerdelijk geselecteerde leptonen uit de hoge MT 2 (ll) regio te verwijderen. Dit is
81
weergegeven in de Figuren 5.6b en 5.7. Daarnaast is het toepassen van deze cut zeer
signaalefficiënt, vermits deze slechts een verlies van 4% oplevert.
7.3
Deel II: Scherpstellen van αISR
in POWHEG+
S
PYTHIA8 voor het modelleren van top quark
paren
Deel II van deze studie situeert zich rond het zoeken naar oplossingen voor een gekend
probleem bij het genereren van tt̄ evenementen m.b.v. de MC generatoren POWHEG+
PYTHIA8. In Fig. 6.1 wordt aangetoond hoe het resultaat van de simulatie in functie van jet-multipliciteit Njets duidelijk afwijkt van wat gemeten wordt in data. Een
oplossing wordt hiertoe voorgesteld waarbij een specifieke parameter in PYTHIA8, nl.
SpaceShower: alphaSvalue, wordt scherpgesteld. Deze wordt in het vervolg van de
tekst aangeduid met αSISR . Het scherpstellen gebeurt d.m.v. Professor, een programma
daartoe speciaal ontworpen. Het scherpstellen wordt in het vervolg tuning genoemd.
Er worden 13 MC runs gegenereerd voor verschillende waarden van αSISR zoals gegeven
in Tabel 6.1. Voor deze runs worden drie tunings uitgevoerd naar een bepaalde observabele (of observabelen) zoals aangeduid in Tabel 6.2. Dit gebeurt aan de hand van een
χ2 methode die in meer detail in Appendix A wordt toegelicht. Elk tuning-resultaat
staat samengevat in Tabel 6.3 met bijhorende onzekerheden.
Het effect van elke tuning poging op de verschillende observabelen wordt weergegeven
in Figuren 6.4, 6.5 en 6.6. Zij worden bovendien vergeleken met de ATLAS ATTBARPOWHEG tune [59], de PYTHIA8 (CUETP8M1) en PYTHIA6 (CTEQ5M) standaarden
in Fig 6.3. Hieruit wordt besloten dat tune 1 de meest optimale is vermits deze met
het ATLAS-resultaat en de PYTHIA6 en PYTHIA8 standaarden in overeenkomst is, de
kleinste onzekerheden heeft en de distributies in Fig. 6.4 accuraat beschrijft.
Voor de optimale tune 1 wordt voor diens onzekerheden nagegaan met welke onzekerheden in de renormalisatieschaal deze corresponderen. Dit wordt grafisch weergegeven
in Fig. 6.7. De onzekerheden corresponderen met een variatie van de renormalisatieparameter volgens de factoren 0.33 en 4.10.
Toepassing van de gevonden tunes op TOP-15-011 distributies [60] is weergegeven in
Figuren 6.9 en 6.10. Hier worden de MC runs voor de verschillende tunes in functie
82
van de observabelen ptT , ∆φtt̄ , ptTt̄ en mtt̄ uitgezet. Ten eerste valt hierbij op dat ptT
uitermate slecht beschreven wordt. Dit is te wijten aan hogere orde effecten die niet in
rekening gebracht zijn en niet opgelost kunnen worden door tuning. Ten tweede valt
op dat de MC uitkomsten voor alle tunes voor de overige distributies erg gelijkaardig
zijn met elkaar. Op het zicht leren we hier dat de beste overeenkomsten toch bij de
hogere waardan van αSISR te vinden zijn. Daarom werden de tt̄ samples voor ICHEP
2016 aangevraagd met de waarde van αSISR = 0.1273, hetgeen dezelfde waarde is van
de PYTHIA6 standaard en dus iets hoger ligt dan de eigen gevonden tunes van orde
0.10-0.11.
Voor een toekomstige poging tot tuning, kan men proberen om αSISR gelijktijdig of opeenvolgend te tunen met de hdamp parameter in POWHEG daar dit gunstige resultaten
levert volgens een studie gedaan door de ATLAS collaboratie [59].
Men kan concluderen dat in deze studie de beschrijving van de Njets observable hersteld
is door het gebruiken van een lagere αSISR waarde dan de standaardwaarde van 0.1365
voor PYTHIA8. Deze aanpassing veroorzaakt geen significante veranderingen voor de
overige top quark distributies (ptT , ∆φtt̄ , ptTt̄ en mtt̄ ) daar deze nog steeds accuraat
beschreven worden door POWHEG+PYTHIA8 behalve ptT . Deze resultaten zijn ook publiek
gemaakt als extra materiaal voor [58].
83
Appendix A
Professor: A tuning tool for Monte
Carlo event generators
Professor (PROcedure For EStimating Systematic errORs) is a parametrisation-based
tuning tool. This means that the approach taken is to parametrise the generator behaviour by fitting a polynomial to the generator response (here referred to as MCb ) for
each observable bin to changes in the P -element parameter vector p = (p1 , ..., pP ). P
denotes the amount of parameters that can be changed in the event-generator.
One uses a polynomial of second order to do the bin parametrisation. Say that p 0 is
a randomly chosen point in parameter space (in between the bounds of a hypercube
determined by the user), then the polynomial has the following form:
(b)
M Cb (p) ≈ α0 +
P
X
i
(b)
βi p0i +
P
X
(b)
γij p0i p0j
(A.1)
i≤j
with p 0 = p - p 0 . The left side of the above equations denotes the output from the
event-generator at parameter point p, whereas the right side contains the unknown
coefficients [α, β, γ]. The amount of coefficients to be determined depends on the order
of the polynomial (here: 2) and the amount of parameters P to be tuned, also referred
to as the dimension of the parameter space. For a 2nd order fit, it can be readily
computed from Eq. A.1:
(P )
N2
= 1 + P + P (P + 1)/2
(A.2)
Once we have a set of functions MCb , we have a very fast way of predicting the gen84
erator behaviour at a certain point in parameter space p, as it is not necessary to run
the generator again. This allows one to construct a goodness of fit function which compares the data to the Monte Carlo, and thus to come up with an optimal point p best
of the tuned parameters, even though p best was not specifically used in the computation.
A.1
Determining the response function MCb(p)
Given sets of sampled points {p} and generator values {M Cb (p)}, eq. A.1 can be transformed into a matrix equation, which is acquired for every bin and for every observable:
e (b)
v(b) = Pc
(A.3)
e contains the variations of the parameter points (p - p 0 ) for different orders. We get
P
in matrix notation, if we presume 2 parameters such that p = (px , py ) and N evaluated
parameter points for a certain bin b, that:
 
α0
  

 βx 
1 x1 y1 x21 x1 y1 y12
v1

 v 2   1 x2 y 2 x2 x2 y 2 y 2  
 βy 
2
2
  

(A.4)
·  
 ..  = 
..
 .  
 γxx 
.

γxy 
2
1 xN yN x2N xN yN yN
vN
{z
}
| {z } |
γyy
e
v (values)
| {z }
P(sampled
parameter sets)
c (coefficients)
with the values vi being the generator computed value. This can be understood if we
simply look at the equation we get for the first row. With x denoting the first parameter
and y denoting the second parameter, we get:
v1 = α0 + x1 βx + y1 βy + x21 γxx + x1 y1 γxy + y12 γyy
(A.5)
with x1 = p0x = px −px0 and y1 = p0y = py −py0 both evaluated at point n◦ 1 in parameter
space. This is exactly eq. A.1.
e known, we still need to compute c(b) . This can be done
With the matrices v(b) and P
e
by a so-called pseudo-inversion Ie of P:
85
e (b)
e P]v
c(b) = I[
(A.6)
Pseudo-inversion is comparable to regular (n×n)-matrix inversion, but here for (m×n)matrices.
A.2
Goodness of Fit (GoF)
In the previous section, the functions MCb (p) were determined, giving us the generator
response for every point in parameter space for every observable bin. What remains
now, is to compute a goodness of fit function that is minimised for a certain parameter
point p best , which can be found from extrapolating the GoF function to a minimum.
It is possible to add per-observable weights wO for each observable O. We choose a
χ2 -function defined according to:
2
χ (p) =
X
O
X (MCb (p) − Rb )2
wO
∆2b
b ∈ O
(A.7)
where Rb is the data value for bin b, and the error ∆b is calculated from the sum
in quadrature of the MC-error and the error in the data. It should be noted that
determination of the weights is a subjective process and should be chosen according to
criteria set by the user.
A.3
Error Estimation: Construction of Eigentunes
Once an optimal set of parameters p best is found, it remains to find the correct errors
on the acquired tunes. The upper and lower bounds we refer to as the eigentunes. For
simplicity we now consider a two dimensional parameter space p = (px ,py ), with optimal tune values (p∗x ,p∗y ). The eigentunes can be constructed by considering a Taylor’s
expansion around the χ2 (p).
2
2
χ (px , py ) ≈ χ
(p∗x , p∗y )
2 2
2 2
1
1
∗ 2∂ χ ∗ 2∂ χ + (px − px )
+ (py − py )
2
∂p2x (p∗x ,p∗y ) 2
∂p2y (p∗ ,p∗ )
x y
2 2 ∂ χ + (px − p∗x )(py − p∗y )
∂px ∂py (p∗x ,p∗y )
86
(A.8)
If we compare this formula to the general equation of a rotated ellipse centred about
(a,b)
(x − a)2 (y − b)2 xy
+
+ 2,
(A.9)
A2
B2
C
with A and B the lengths of the axes and B14 − 4 A12 C12 < 01 , we learn that for a specific
value of ∆χ2 = χ2 (px , py ) - χ2 (p∗x , p∗y ) an ellipse in parameter space can be drawn.
The standard Professor prescription is to choose ∆χ2 = χ2 (p∗x , p∗y ). This means the
minimum of the χ2 is shifted up by the value of χ2 (p∗x , p∗y ). The outer edges of the
ellipse are then taken to be the errors on the tunes as depicted in Fig. A.1
1=
Figure A.1: Ellipse denoting equal values of the χ2 in parameter space.
The ellipse of equal χ2 is rotated because of the fact that the parameters px and py
are correlated.
This follows from Eq. A.7 in which it is clear that the term containing
∂ 2 χ2 becomes zero if a variation of the χ2 in p1 does not depend on a variation
∂px ∂py (p∗x ,p∗y )
in p2 (or vice versa) and would then mean that the ellipse has principal axes parallel
to the x- and y-axis.
Write the ellipse equation like y12 = αt2 + βt + γ with t = xy , and realise that αt2 + βt + γ can never
be equal to 0, because y12 is always positive. This requires a negative discriminant D = β 2 − 4αγ.
1
87
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